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 Communications in Combinatorics and OptimizationNumber of Followers: 0     Open Access journal ISSN (Print) 2538-2128 - ISSN (Online) 2538-2136 Published by Azarbaijan Shahide Madani University  [1 journal]
• Some properties of the essential annihilating-ideal graph of commutative
rings

• Abstract: Let $\mathcal{S}$ be a commutative ring with unity and $A(\mathcal{S})$ denotes the set of annihilating-ideals of $\mathcal{S}$. The essential annihilating-ideal graph of $\mathcal{S}$, denoted by $\mathcal{EG}(\mathcal{S})$, is an undirected graph with $A^*(\mathcal{S})$ as the set of vertices and &nbsp; for distinct $\mathcal{I}, \mathcal{J} \in A^*(\mathcal{S})$, $\mathcal{I} \sim \mathcal{J}$ is an edge if and only if $Ann(\mathcal{IJ}) \leq_e \mathcal{S}$. In this paper, we classify the Artinian rings $\mathcal{S}$ for which $\mathcal{EG}(\mathcal{S})$ is projective. We also discuss the coloring of $\mathcal{EG}(\mathcal{S})$. Moreover, we discuss the domination number of $\mathcal{EG}(\mathcal{S})$.

• Roman domination number of signed graphs

• Abstract: A function $f:V\rightarrow \{0,1,2\}$ on a signed graph $S=(G,\sigma)$ is a \textit{Roman dominating function(RDF)} if $f(N[v]) = f(v) + \sum_{u \in N(v)} \sigma(uv)f(u) \geq 1$ for all $v\in V$ and for each vertex $v$ with $f(v)=0$ there is a vertex $u$ in $N^+(v)$ such that $f(u) = 2$. The weight of an RDF $f$ is given by $\omega(f) = \sum_{v\in V}f(v)$ and the minimum weight among all the $RDF$s on $S$ is called the Roman domination number $\gamma_R(S)$. Any RDF on $S$ with the minimum weight is known as a $\gamma_R(S)$-function. In this article we obtain certain bounds for $\gamma_{R}$ and characterise the signed graphs attaining small values for $\gamma_R.$

• Uniqueness of rectangularly dualizable graphs

• Abstract: A generic rectangular partition &nbsp; is a partition of a rectangle into a finite number of rectangles provided that no four of them meet at a point. &nbsp;A graph $\mathcal{H}$ is called &nbsp;dual of a plane graph $\mathcal{G}$ if there is one$-$to$-$one correspondence between the vertices of $\mathcal{G}$ and the regions of $\mathcal{H}$, and &nbsp;two vertices of $\mathcal{G}$ are adjacent if and only if the corresponding regions of $\mathcal{H}$ are adjacent. A plane graph is a&nbsp; rectangularly dualizable graph &nbsp;if its dual &nbsp;can be embedded as a &nbsp;rectangular partition. &nbsp; A rectangular dual&nbsp; $\mathcal{R}$ of a plane graph $\mathcal{G}$ is a partition of a &nbsp;rectangle &nbsp;into $n-$rectangles such that (i) no four rectangles of $\mathcal{R}$ meet at a point, (ii) &nbsp;rectangles in &nbsp;$\mathcal{R}$ are mapped to vertices of $\mathcal{G}$, &nbsp;and (iii) &nbsp;two rectangles in $\mathcal{R}$ share a common boundary segment if and only if the corresponding vertices are adjacent in $\mathcal{G}$. In this paper, we derive a necessary and sufficient &nbsp;for a rectangularly dualizable graph $\mathcal{G}$ to admit a unique rectangular dual upto combinatorial &nbsp;equivalence. Further we show that $\mathcal{G}$ always admits &nbsp; a slicible as well as an area$-$universal &nbsp;rectangular dual.

• Bounds of point-set domination number

• Abstract: A subset $D$ of the vertex set $V(G)$ in a graph $G$ is a point-set dominating set (or, in short, psd-set) of $G$ if for every set $S\subseteq V- D$, there exists a vertex $v\in D$ such that the induced subgraph $\langle S\cup \{v\}\rangle$ is connected. &nbsp;The minimum cardinality of a psd-set of $G$ is called the point-set domination number of $G$. In this paper, we establish two sharp lower bounds for point-set domination number of a graph in terms of its diameter and girth. We characterize graphs for which lower bound of point set domination number is attained in terms of its diameter. We also establish an upper bound and give some classes of graphs which attains the upper bound of point set domination number.

• Coalition Graphs

• Abstract: A coalition in a graph $G = (V, E)$ consists of two disjoint sets $V_1$ and $V_2$ of vertices, such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1 \cup V_2$ is a dominating set of $G$. A coalition partition in a graph $G$ of order $n = V$ is a vertex partition $\pi = {V_1, V_2, \ldots, V_k}$ such that every set $V_i$ either is a dominating set consisting of a single vertex of degree $n-1$, or is not a dominating set but forms a coalition with another set $V_j$. Associated with every coalition partition $\pi$ of a graph $G$ is a graph called the coalition graph of $G$ with respect to $\pi$, denoted $CG(G,\pi)$, the vertices of which correspond one-to-one with the sets $V_1, V_2, \ldots, V_k$ of $\pi$ and two vertices are adjacent in $CG(G,\pi)$ if and only if their corresponding sets in $\pi$ form a coalition. In this paper, we initiate the study of coalition graphs and we show that every graph is a coalition graph.

• On signs of several Toeplitz--Hessenberg determinants whose elements
contain central Delannoy numbers

• Abstract: In the paper, by virtue of Wronski&#039;s formula and Kaluza&#039;s theorem for the power series and its reciprocal, and with the aid of the logarithmic convexity of a sequence constituted by central Delannoy numbers, the authors present negativity of several Toeplitz--Hessenberg determinants whose elements contain central Delannoy numbers and combinatorial numbers.

• Cycle transit function and betweenness

• Abstract: Transit functions are introduced to study betweenness, intervals and convexity in an axiomatic setup on graphs and other discrete structures. Prime example of a transit function on graphs is the well studied interval function of a connected graph. In this paper, we study the Cycle transit function $\mathcal{C}( u,v)$ on graphs which is a transit function derived from the interval function. We study the betweenness properties and also characterize graphs in which the cycle transit function coincides with the interval function. We also characterize graphs where $\mathcal{C}( u,v)\cap \mathcal{C}( v,w) \cap \mathcal{C}( u,w) \le 1$ as an analogue of median graphs.

• On several new closed-form evaluations for the generalized hypergeometric
functions

• Abstract: The main objective of this paper is to establish as many as thirty new closed-form evaluations of the generalized hypergeometric function $_{q+1}F_q(z)$ for $q= 2, 3$. This is achieved by means of separating the generalized hypergeometric function $_{q+1}F_q(z)$ for $q=1, 2, 3$ into even and odd components together with the use of several known infinite series involving reciprocal of the non-central binomial coefficients obtained earlier by L. Zhang and W. Ji.

• On local antimagic chromatic number of various join graphs

• Abstract: A local antimagic edge labeling of a graph $G=(V,E)$ is a bijection $f:E\rightarrow\{1,2,\dots, E \}$ such that the induced vertex labeling $f^+:V\rightarrow \mathbb{Z}$ given by $f^+(u)=\sum f(e),$ where the summation runs over all edges $e$ incident to $u,$ has the property that any two adjacent vertices have distinct labels. A graph $G$ is said to be locally antimagic if it admits a local antimagic edge labeling. The local antimagic chromatic number $\chi_{la}(G)$ is the minimum number of distinct induced vertex labels over all local antimagic &nbsp;labelings of $G.$ In this paper we obtain sufficient conditions under which $\chi_{la}(G\vee H),$ where $H$ is either a cycle or the empty graph $O_n=\overline{K_n},$ satisfies a sharp upper bound. Using this we determine the value of $\chi_{la}(G\vee H)$ for many wheel related graphs $G.$

• A homogeneous predictor-corrector algorithm for stochastic nonsymmetric
convex conic optimization with discrete support

• Abstract: We consider a stochastic convex optimization problem over nonsymmetric cones with discrete support. This class of optimization problems has not been studied yet. By using a logarithmically homogeneous self-concordant barrier function, we present a homogeneous predictor-corrector interior-point algorithm for solving stochastic nonsymmetric conic optimization problems. We also derive an iteration bound for the proposed algorithm. Our main result is that we uniquely combine a nonsymmetric algorithm with efficient methods for computing the predictor and corrector directions. Finally, we describe a realistic application and present computational results for instances of the stochastic facility location problem formulated as a stochastic nonsymmetric convex conic optimization problem.

• Total restrained Roman domination

• Abstract: Let $G$ be a graph with vertex set $V(G)$. A Roman dominating function &nbsp;(RDF) on a graph $G$ is a function $f:V(G)\longrightarrow\{0,1,2\}$ such that every vertex $v$ &nbsp;with $f(v)=0$ is adjacent to a vertex &nbsp;$u$ with $f(u)=2$. If $f$ is an RDF on $G$, then let $V_i=\{v\in V(G): f(v)=i\}$ for $i\in\{0,1,2\}$. An RDF $f$ is called a restrained (total) Roman dominating function if the subgraph induced by $V_0$ &nbsp;(induced by $V_1\cup V_2$) has no isolated vertex. A total and restrained Roman dominating function is a total restrained Roman &nbsp;dominating function. &nbsp;The total restrained Roman domination number $\gamma_{trR}(G)$ on a graph $G$ is the minimum weight of a total restrained Roman dominating function on the graph $G$.We initiate the study of total restrained Roman domination number and present several sharp bounds on $\gamma_{trR}G)$. In addition, we determine this parameter for some classes of graphs.

• Linear-time construction of floor plans for plane triangulations

• Abstract: This paper focuses on a novel approach for producing a floor plan (FP), either a rectangular (RFP) or an orthogonal (OFP) based on the concept of orthogonal drawings, which satisfies the adjacency relations given by any bi-connected plane triangulation G.Previous algorithms for constructing a FP are primarily restricted to the cases given below:(i) A bi-connected plane triangulation without separating triangles (STs) and with at most 4 corner implying paths (CIPs), known as properly triangulated planar graph (PTPG).(ii) A bi-connected plane triangulation with an exterior face of length 3 and no CIPs, known as maximal planar graph (MPG).The FP obtained in the above two cases is a RFP or an OFP respectively. In this paper, we present the construction of a FP (RFP if exists, else an OFP), for a bi-connected plane triangulation G in linear-time.

• The Cartesian product of wheel graph and path graph is antimagic

• Abstract: Suppose each edge of a simple connected undirected graph is given a unique number from the numbers $1, 2, \dots,$q$, where$q$is the number of edges of that graph. Then each vertex is labelled with sum of the labels of the edges incident to it. If no two vertices have the same label, then the graph is called an antimagic graph. We prove that the Cartesian product of wheel graph and path graph is antimagic. • Double Roman domination in graphs: algorithmic complexity • Abstract: Let$G=(V,E)$be a graph. &nbsp;A double Roman dominating function &nbsp;(DRDF) of &nbsp;$G $&nbsp; is a function &nbsp;$f:V\to \{0,1,2,3\}$&nbsp;such that, for each$v\in V$with$f(v)=0$, &nbsp;there is a vertex$u $&nbsp;adjacent to$v$&nbsp;with$f(u)=3$or there are vertices$x$and$y $&nbsp;adjacent to$v$&nbsp;such that &nbsp;$f(x)=f(y)=2$and for each$v\in V$with$f(v)=1$, &nbsp;there is a vertex$u $&nbsp; &nbsp;adjacent to$v$&nbsp; &nbsp;with &nbsp;$f(u)&gt;1$. &nbsp;The weight of a DRDF$f$is &nbsp;$ f (V) =\sum_{ v\in V} f (v)$. &nbsp; Let$n$and &nbsp;$k$be integers such that &nbsp;$3\leq 2k+ 1 &nbsp;\leq n$. &nbsp;The &nbsp; generalized Petersen graph$GP (n, k)=(V,E) $&nbsp;is the &nbsp;graph &nbsp;with &nbsp;$V=\{u_1, u_2,\ldots, &nbsp;u_n\}\cup\{v_1, v_2,\ldots, v_n\}$and$E=\{u_iu_{i+1}, u_iv_i, v_iv_{i+k}: &nbsp;1 \leq i \leq n\}$, where &nbsp;addition is taken &nbsp;modulo$n$. In this paper, &nbsp;we firstly &nbsp; prove that the &nbsp;decision &nbsp; &nbsp; problem &nbsp;associated with &nbsp; double Roman domination is NP-omplete even restricted to planar bipartite graphs with maximum degree at most 4. &nbsp;Next, we &nbsp; give &nbsp;a dynamic programming algorithm for &nbsp;computing a minimum DRDF (i.e., a &nbsp;DRDF &nbsp; with minimum weight &nbsp;along &nbsp;all &nbsp; DRDFs) &nbsp;of$GP(n,k )$&nbsp;in$O(n81^k)$time and space &nbsp;and so a &nbsp;minimum DRDF &nbsp;of$GP(n,O(1))$&nbsp;can be computed in$O( n)$time and space. • Signless Laplacian eigenvalues of the zero divisor graph associated to finite commutative ring$ \mathbb{Z}_{p^{M_{1}}~q^{M_{2}}} $• Abstract: For a commutative ring$R$with identity$1\neq 0$, let the set$Z(R)$denote the set of zero-ivisors and let$Z^{*}(R)=Z(R)\setminus \{0\}$be the set of non-zero zero-divisors of$R$. &nbsp;The zero-divisor graph of$R$, denoted by$\Gamma(R)$, is a simple graph whose vertex set is$Z^{*} (R)$and two vertices$u, v \in Z^*(R)$are adjacent if and only if$uv=vu=0$. In this article, we find the signless Laplacian spectrum of the zero divisor graphs$ \Gamma(\mathbb{Z}_{n}) $for$ n=p^{M_{1}}q^{M_{2}}$, where$ p&lt;q $are primes and$ M_{1} , M_{2} $are positive integers. • On Sombor coindex of graphs • Abstract: In this paper, we explore several properties of Sombor coindex of a finite simple graph and we derive a bound for the total Sombor index. We also explore its relations to the Sombor index, the Zagreb coindices, forgotten coindex and other important graph parameters. We further compute the bounds of the Somber coindex of some graph operations and derived explicit formulae of Sombor coindex for some well-known graphs as application. • An upper bound on triple Roman domination • Abstract: For a graph$G=(V,E)$, a triple Roman dominating function (3RD-function) is a function$f:V\to \{0,1,2,3,4\}$having the property that (i) if$f(v)=0$then$v$must have either one neighbor$u$with$f(u)=4$, or two neighbors$u,w$with$f(u)+f(w)\ge 5$or three neighbors$u,w,z$with$f(u)=f(w)=f(z)=2$, (ii) if$f(v)=1$then$v$must have one neighbor$u$with$f(u)\ge 3$or two neighbors$u,w$with$f(u)=f(w)=2$, and (iii) if$f(v)=2$then$v$must have one neighbor$u$with$f(u)\ge 2$. The weight of a 3RDF$f$is the sum$f(V)=\sum_{v\in V} f(v)$, and the minimum weight of a 3RD-function on$G$is the triple Roman domination number of$G$, denoted by$\gamma_{[3R]}(G)$. In this paper, we prove that for any connected graph$G$of order$n$with minimum degree at least two,&nbsp;$\gamma_{[3R]}(G)\leq \frac{3n}{2}$. • Domination parameters of the splitting graph of a graph • Abstract: Let$G=(V,E)$be a graph of order$n$and size$m.$The graph$Sp(G)$obtained from$G$by adding a new vertex$v'$for every vertex$v\in V$and joining$v'$to all neighbors of$v$in$G$is called the splitting graph of$G.$In this paper, we determine the domination number, the total domination number, connected domination number, paired domination number and independent domination number for the splitting graph$Sp(G).$• On the Total Monophonic Number of a Graph • Abstract: Let G = (V,E) be a connected graph of order n. A path P in G which does not have a chord is called a monophonic path. A subset S of V is called a monophonic set if every vertex v in V lies in a x-y monophonic path where x, y 2 S. If further the induced subgraph G[S] has no isolated vertices, then S is called a total monophonic set. The total monophonic number mt(G) and the upper total monophonic number m+t (G) are respectively the minimum cardinality of a total monophonic set and the maximum cardinality of a minimal total monophonic set. In this paper we determine the value of these parameters for some classes of graphs and establish bounds for the same. We also prove the existence of graphs with prescribed values for mt(G) and m+t (G). • Weak Roman domination stable graphs upon edge-addition • Abstract: A Roman dominating function (RDF) on a graph$G$is a function$f: V(G) \to \{0, 1, 2\}$such that every vertex with label 0 has a neighbor with label 2. A vertex$u$with$f(u)=0$is said to be undefended if it is not adjacent to a vertex with$f(v)&gt;0$. The function$f:V(G) \to \{0, 1, 2\}$is a weak Roman dominating function (WRDF) if each vertex$u$with$f(u) = 0$is adjacent to a vertex$v$with$f(v) &gt; 0$such that the function$f^{\prime}: V(G) \to \{0, 1, 2\}$defined by$f^{\prime}(u) = 1$,$f^{\prime}(v) = f(v) - 1$and$f^{\prime}(w) = f(w)$if$w \in V - \{u, v\}$, has no undefended vertex. A graph$G$is said to be Roman domination stable upon edge addition, or just$\gamma_R$-EA-stable, if$\gamma_R(G+e)= \gamma_R(G)$for any edge$e \notin E(G)$. We extend this concept to a weak Roman dominating function as follows: A graph$G$is said to be weak Roman domination stable upon edge addition, or just$\gamma_r$-EA-stable, if$\gamma_r(G+e)= \gamma_r(G)$for any edge$e \notin E(G)$. In this paper, we study$\gamma_r$-EA-stable graphs, obtain bounds for &nbsp;$\gamma_r$-EA-stable graphs and&nbsp; characterize$\gamma_r$-EA-stable trees which attain the bound.&nbsp; • Signed total Italian domination in digraphs • Abstract: Let$D$be a finite and simple digraph with vertex set$V(D)$. A signed total Italian dominating function (STIDF) on a digraph$D$is a function$f:V(D)\rightarrow\{-1,1,2\}$satisfying the conditions that (i)$\sum_{x\in N^-(v)}f(x)\ge 1$for each$v\in V(D)$, where$N^-(v)$consists of all vertices of$D$from which arcs go into$v$, and (ii) every vertex$u$for which$f(u)=-1$has an in-neighbor$v$for which$f(v)=2$or two in-neighbors$w$and$z$with$f(w)=f(z)=1$. The weight of an &nbsp;STIDF$f$is$\sum_{v\in V(D)}f(v)$. The signed total Italian domination number$\gamma_{stI}(D)$of$D$is the minimum weight of an STIDF on$D$. In this paper we initiate the study of the signed total Italian domination number of digraphs, and we &nbsp;present different bounds on$\gamma_{stI}(D)$. In addition, we determine the signed total Italian domination number of some classes of digraphs. • Outer-independent total 2-rainbow dominating function of graphs • Abstract: Let$G=(V,E)$be a simple graph with vertex set$V$and edge set$E$. An {outer-independent total$2$-rainbow dominating function of a graph$G$is a function$f$from$V(G)$to the set of all subsets of$\{1,2\}$such that the following conditions hold: (i) for any vertex$v$with$f(v)=\emptyset$we have$\bigcup_{u\in N_G(v)} f(u)=\{1,2\}$, (ii) the set of all vertices$v\in V(G)$with$f(v)=\emptyset$is independent and (iii)$\{v\mid f(v)\neq\emptyset\}$has no isolated vertex. The outer-independent total$2$-rainbow domination number of$G$, denoted by${\gamma}_{oitr2}(G)$, is the minimum value of$\omega(f)=\sum_{v\in V(G)} f(v) $over all such functions$f$. In this paper, we study the outer-independent total$2$-rainbow domination number of$G$and classify all graphs with outer-independent total$2$-ainbow domination number belonging to the set$\{2,3,n\}$. Among other results, we present some sharp bounds concerning the invariant. • Normalized distance Laplacian matrices for signed graphs • Abstract: In this paper, we introduce the notion of normalized distance Laplacian matrices for signed graphs corresponding to the two signed distances defined for signed graphs. We characterize balance in signed graphs using these matrices and compare the normalized distance Laplacian spectral radius of signed graphs with that of all-negative signed graphs. Also we characterize the signed graphs having maximum normalized distance Laplacian spectral radius. • 2S3 transformation for Dyadic fractions in the interval (0, 1) • Abstract: The$2S3$transformation, which was first described for positive integers, has been defined for dyadic rational numbers in the open interval$(0,1)$&nbsp;in this study. &nbsp;The set of dyadic rational numbers &nbsp;is a Pr&uuml;fer 2-group. For the dyadic$2S3$transformation$T_{ds}(x)$, the restricted multiplicative and additive properties have been established. Graph parameters are used to generate more combinatorial outcomes for these properties. The relationship between the SM dyadic sum graph's automorphism group and the symmetric group has been investigated. • A Study on Graph Topology • Abstract: The concept of topology defined on a set can be extended to the field of graph theory by defining the notion of graph topologies on graphs where we consider a collection of subgraphs of a graph$G$in such a way that this collection satisfies the three conditions stated similarly to that of the three axioms of point-set topology. This paper discusses an introduction and basic concepts to the graph topology. A subgraph of$G$is said to be open if it is in the graph topology$sT_G$. The paper also introduces the concept of the closed graph and the closure of graph topology in graph topological space using the ideas of decomposition-complement and neighborhood-complement. • A note on the lower bound of the second Zagreb index of trees with a given Roman domination number • Abstract: For a (molecular) graph, the second Zagreb index$M_2(G)$is equal to the sum of the products of the degrees of pairs of adjacent vertices. Roman dominating function$RDF$of$G$is a function$f:V(G)rightarrow {0,1,2}$satisfying the condition that every vertex with label 0 is adjacent to a vertex with label 2. The weight of an$RDFf$is$w(f)=sum_{vin V(G)} f(v)$. The Roman domination number of$G$, denoted by$gamma_R (G)$, is the minimum weight among all RDF in$G$. In this paper, we present a lower bound on the second Zagreb index of trees with$n$vertices and Roman domination number and thus settle one problem given in [On the Zagreb indices of graphs with given Roman domination number, Commun. Comb. Optim. DOI : 10.22049/CCO.2021.27439.1263 (article in press)]. • More on the bounds for the skew Laplacian energy of weighted digraphs • Abstract: Let$\mathscr{D}$be a simple connected digraph with$n$vertices and$m$arcs and let$W(\mathscr{D})=\mathscr{D},w)$be the weighted digraph corresponding to$\mathscr{D}$, where the weights are taken from the set of non-zero real numbers. Let$nu_1,nu_2, \dots,nu_n$be the eigenvalues of the skew Laplacian weighted matrix$\widetilde{SL}W(\mathscr{D})$of the weighted digraph$W(\mathscr{D})$. In this paper, we discuss the skew Laplacian energy$\widetilde{SLE}W(\mathscr{D})$of weighted digraphs and obtain the skew Laplacian energy of the weighted star$W(\mathscr{K}_{1, n})$for some fixed orientation to the weighted arcs. We obtain lower and upper bounds for$\widetilde{SLE}W(\mathscr{D})$and show the existence of weighted digraphs attaining these bounds.&nbsp; • Roman domination in signed graphs • Abstract: Let$S = (G,\sigma)$be a signed graph. A function$f: V \rightarrow \{0,1,2\}$is a Roman dominating function on$S$if$(i)$for each$v \in V,f(N[v]) = f(v) + \sum_{u \in N(v)} \sigma(uv ) f(u) \geq 1$and$(ii)$for each vertex$ v $with$ f(v) = 0 $there exists a vertex$u \in N^+(v)$such that$f(u) = 2.$In this paper we initiate a study on Roman dominating function on signed graphs. We characterise the signed paths, cycles and stars that admit a Roman dominating function. • Cop-edge critical generalized Petersen and Paley graphs • Abstract: Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. We study textit{cop-edge critical} graphs, i.e. graphs$G$such that for any edge$e$in$E(G)$either$c(G-e)&lt; c(G)$or$c(G-e)&gt;c(G)$. In this article, we study the edge criticality of generalized Petersen graphs and Paley graphs.&nbsp; • Unit$\mathbb{Z}_q$-Simplex codes of type α and zero divisor$\mathbb{Z}_q$-Simplex codes • Abstract: In this paper, we have punctured unit$\mathbb{Z}_q$-Simplex code &nbsp;and constructed a new code called unit$\mathbb{Z}_q$-Simplex code of type$\alpha$. In particular, we find the parameters of &nbsp;these codes and have proved that it is an$\left[\phi(q)+2, ~\hspace{2pt} 2, ~\hspace{2pt} \phi(q)+2 - \frac{\phi(q)}{\phi(p)}\right]\mathbb{Z}_q$-linear code$\text{if} ~ k=2$and$\left[\frac{\phi(q)^k-1}{\phi(q)-1}+\phi(q)^{k-2}, ~k,~ \frac{\phi(q)^k-1} {\phi(q)-1}+\phi(q)^{k-2}-\left(\frac{\phi(q)}{\phi(p)}\right)\left(\frac{\phi(q)^{k-1}-1}{\phi(q)-1}+\phi(q)^{k- 3}\right)\right]\mathbb{Z}_q$-linear code if$k \geq 3, $where$p$is the smallest prime divisor of$q.$&nbsp; For$q$is a prime power and rank$k=3,$we have given the &nbsp;weight distribution of &nbsp;unit$\mathbb{Z}_q$-Simplex codes&nbsp; of type$\alpha$. Also, we have introduced some new code from &nbsp;$\mathbb{Z}_q$-Simplex code called zero divisor$\mathbb{Z}_q$-Simplex code and proved that it is an$\left[ \frac{\rho^k-1}{\rho-1}, \hspace{2pt} k, \hspace{2pt} \frac{\rho^k-1}{\rho-1}-\left(\frac{\rho^{(k-1)}-1}{\rho-1}\right)\left(\frac{q}{p}\right) \right]\mathbb{Z}_{q}$-linear code, where$\rho = q-\phi(q)$and$p$is the smallest prime divisor of$q.$Further, we obtain &nbsp;weight distribution of&nbsp; zero divisor$\mathbb{Z}_q$-Simplex code for rank$k=3$and$q$is a prime power. • Line signed graph of a signed unit graph of commutative rings • Abstract: In this paper we characterize the commutative rings with unity for which line signed graph of signed unit graph is balanced and consistent. To do this, first we derive some sufficient conditions for balance and consistency of signed unit graphs. The results have been demonstrated with ample number of examples. • A survey on the Intersection graphs of ideals of rings • Abstract: Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a non-zero intersection. The ideal structure of a ring reflects many ring theoretical properties. Thus much research has been conducted last few years to explore the properties of G(R). This is a survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009. • On the total liar's domination of graphs • Abstract: For a graph$G$, a set$L$of vertices is called a total liar's domination if$ N_G(u)\cap L \geq 2$for any$u\in V(G)$and$ (N_G(u)\cup N_G(v))\cap L \geq 3$for any distinct vertices$u,v\in V(G)$. The total liar&rsquo;s domination number is the cardinality of a minimum total liar&rsquo;sdominating set of$G$and is denoted by$\gamma_{TLR}(G)$. In this paper we study the total liar's domination numbers of join and products of graphs. • Complexity of the paired domination subdivision problem • Abstract: The paired domination subdivision number of a graph$G$is the minimum number of edges that must be subdivided (where each edge in$G$can be subdivided at most once) in order to increase the paired domination number of$G$. In this note, we show that the problem of computing the paired-domination subdivision number is NP-hard for bipartite graphs. • Algorithmic aspects of total Roman$\{2\}$-domination in graphs • Abstract: For a simple, undirected, connected graph$G$, a function$h : V \rightarrow \lbrace 0, 1, 2 \rbrace$is called a total Roman$\{2\}$-dominating function (TR2DF) if for every vertex$v$in$V$with weight$0$, either there exists a vertex$u$in$N_G(v)$with weight$2$, or at least two vertices$x, y$in$N_G(v)$each with weight$1$, and the subgraph induced by the vertices with weight more than zero has no isolated vertices. The weight of TR2DF$h$is$\sum_{p \in V} h(p)$. The problem of determining &nbsp;TR2DF of minimum weight is called minimum total Roman \{2\}-domination problem (MTR2DP). We show that MTR2DP is polynomial time solvable for bounded treewidth graphs, threshold graphs and chain graphs. We design a$2 (\ln(\Delta - 0.5) + 1.5)$-approximation algorithm for the MTR2DP and show that the same cannot have$(1 - \delta) \ln V $ratio approximation algorithm &nbsp;for any$\delta &gt; 0$unless$P = NP$. &nbsp;Next, we show that MTR2DP is APX-hard for graphs with$ \Delta=4$. &nbsp;Finally, we show that the domination and TR2DF problems are not equivalent in computational complexity aspects. • Eccentric completion of a graph • Abstract: The eccentric graph$G_e$of a graph$G$is a derived graph with the vertex set same as that of$G$and two vertices in$G_e$are adjacent if one of them is the eccentric vertex of the other. In this paper, the concepts of iterated eccentric graphs and eccentric completion of a graph are introduced and discussed. • Inverse problem for the Forgotten and the hyper Zagreb indices of trees • Abstract: Let$G=(E(G),V(G))$be a (molecular) graph with vertex set$V(G)$and edge set$E(G)$. The forgotten Zagreb index and the hyper Zagreb index of G are defined by$F(G) = \sum_{u \in V(G)} d(u)^{3}$and$HM(G) = \sum_{uv \in E(G)}(d(u)+d(v))^{2}$where$d(u)$and d(v) are the degrees of the vertices$u$and$v$in$G$, respectively. A recent problem called the inverse problem deals with the numerical realizations of topological indices. We see that there exist trees for all even positive integers with$F(G)&gt;88$and with$HM(G)&gt;158$. Along with the result, we show that there exist no trees with$F(G) &lt; 90$and$HM(G) &lt; 160$with some exceptional even positive integers and hence characterize the forgotten Zagreb index and the hyper Zagreb index for trees. • Extreme outer connected monophonic graphs • Abstract: For a connected graph$G$of order at least two, a&nbsp; set$S$of vertices in a graph$G$is said to be an \textit{outer connected monophonic set} if$S$is a monophonic set of$G$and either$S=V$or the subgraph induced by$V-S$is connected. The minimum cardinality of an outer connected monophonic set of$G$is the \textit{outer connected monophonic number} of$G$and is denoted by$m_{oc}(G)$.&nbsp; The number of extreme vertices in$G$is its \textit{extreme order}$ex(G)$.&nbsp; A graph$G$is said to be an \textit{extreme outer connected monophonic graph} if$m_{oc}(G)$=$ex(G)$.&nbsp; Extreme outer connected monophonic graphs of order$p$with outer connected monophonic number$p$and extreme outer connected monophonic graphs of order$p$with outer connected monophonic number$p-1$are characterized.&nbsp; It is shown that for every pair$a, b$of integers with$0 \leq a \leq b$and$b \geq 2$, there exists a connected graph$G$with$ex(G) = a$and$m_{oc}(G) = b$. Also, it is shown that&nbsp; for positive integers$r,d$and$k \geq 2$with$r &lt; d$, there exists an extreme outer connected monophonic graph$G$with monophonic radius$r$, monophonic diameter$d$and outer connected monophonic number$k$. • Entire Wiener index of graphs • Abstract: Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and bonds by the edges of this graph. Topological graph indices have been successfully used in determining the structural properties and in predicting certain physicochemical properties of chemical compounds. Wiener index is the oldest topological index which can be used for analyzing intrinsic properties of a molecular structure in chemistry. The Wiener index of a graph$G$is equal to the sum of distances between all pairs of vertices of$G$. Recently, the entire versions of several indices have been introduced and studied due to their applications. Here we introduce the entire Wiener index of a graph. Exact values of this index for trees and some graph families are obtained, some properties and bounds for the entire Wiener index are established. Exact values of this new index for subdivision and$k$-subdivision graphs and some graph operations are obtained. • A new upper bound on the independent$2$-rainbow domination number in trees • Abstract: A$2$-rainbow dominating function on a graph$G$is a function$g$that assigns to each vertex a set of colors chosen from the subsets of$\{1, 2\}$so that for each vertex with$g(v) =\emptyset$we have$\bigcup_{u\in N(v)} g(u) = \{1, 2\}$. The weight of a$2$-rainbow dominating function$g$is the value$w(g) = \sum_{v\in V(G)} f(v) $. A$2$-rainbow dominating function$g$is an independent$2$-rainbow dominating function if no pair of vertices assigned nonempty sets are adjacent. The$2$-rainbow domination number$\gamma_{r2}(G)$(respectively, the independent$2$-rainbow domination number$i_{r2}(G)$) is the minimum weight of a$2$-rainbow dominating function (respectively, independent$2$-rainbow dominating function) on$G$. We prove that for any tree$T$of order$n\geq 3$, with$\ell$leaves and$s$support vertices,$i_{r2}(T)\leq (14n+\ell+s)/20$, thus improving the bound given in [Independent 2-rainbow domination in trees, Asian-Eur. J. Math. 8 (2015) 1550035] under certain conditions. • Algorithmic aspects of certified domination in graphs • Abstract: A dominating set$ D $of a graph$ G=(V,E) $is called a certified dominating set of$ G $if$\vert N(v) \cap (V \setminus D)\vert$is either 0 or at least 2 for all$ v \in D$. The certified domination number$\gamma_{cer}(G) $is the minimum cardinality of a certified dominating set of$ G $. In this paper, we prove that the decision problem corresponding to$\gamma_{cer}(G) $is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. We also prove that it is linear time solvable for chain graphs, threshold graphs and bounded tree-width graphs. • Regular graphs with large Italian domatic number • Abstract: For a graph$G$, an Italian dominating function is a function$f: V(G) \rightarrow \{0,1,2\}$such that for each vertex$v \in V(G)$either$f(v) \neq 0$, or$\sum_{u \in N(v)} f(u) \geq 2$. If a family$\mathcal{F} = \{f_1, f_2, \dots, f_t\}$of distinct Italian dominating functions satisfy$\sum^t_{i = 1} f_i(v) \leq 2$for each vertex$v$, then this is called an Italian dominating family. In [L. Volkmann, The {R}oman {$\{2\}$}-domatic number of graphs, Discrete Appl. Math.&nbsp; 258 (2019), 235--241], Volkmann defined the Italian domatic number of$G$,$d_{I}(G)$, as the maximum cardinality of any Italian dominating family. In this same paper, questions were raised about the Italian domatic number of regular graphs. In this paper, we show that two of the conjectures are false, and examine some exceptions to a Nordhaus-Gaddum type inequality. • A note on Roman$k$-tuple domination number • Abstract: &lrm;&lrm;For an integer$k\geq 2$&lrm;, &lrm;a Roman$k$-tuple dominating function&lrm;, &lrm;(or just RkDF)&lrm;, &lrm;in a graph$G$is a function$f \colon V(G) \rightarrow \{0&lrm;, &lrm;1&lrm;, &lrm;2\}$satisfying the condition that every vertex$u$for which$f(u) = 0$is adjacent to at least$k$vertices$v$for which$f(v) = 2$&lrm;, &lrm;and every vertex$u$for which$f(u) \neq 0$is adjacent to at least$k-1$vertices$v$for which$f(v) = 2$&lrm;. &lrm;The Roman$k$-tuple domination number of &lrm;$&lrm;G&lrm;$&lrm;&lrm; &lrm;is the minimum weight of an RkDF in$G$. &lrm;In this note we settle two problems posed in [Roman$k$-tuple Domination in Graphs&lrm;, &lrm;Iranian J&lrm;. &lrm;Math&lrm;. &lrm;Sci&lrm;. &lrm;Inform&lrm;. &lrm;15 (2020)&lrm;, &lrm;101--115]&lrm;. • Terminal status of vertices and terminal status connectivity indices of graphs with its applications to properties of cycloalkanes • Abstract: In this article the terminal status of a vertex and terminal status connectivity indices of a connected graph have introduced. Explicit formulae for the terminal status of vertices and for terminal status connectivity indices of certain graphs are obtained. Also some bounds are given for these indices. Further these indices are used for predicting the physico-chemical properties of cycloalkanes and it is observed that the correlation of physico-chemical properties of cycloalkanes with newly introduced indices is better than the correlation with other indices. • Enumeration of k-noncrossing trees and forests • Abstract: A &nbsp;$k$-noncrossing tree is a noncrossing tree where each node receives a label in$\{1,2,\ldots,k\}$such that the sum of labels along an ascent does not exceed$k+1,$if we consider a path from a fixed vertex called the root. In this paper, we provide a proof for a formula that counts the number of$k$-noncrossing trees in which the root (labelled by$k$) has degree$d$. We also find a formula for the number of forests in which each component is a$k$-noncrossing tree whose root is labelled by$k$. • New bounds on Sombor index • Abstract: The Sombor index of the graph$G$is a degree based topological index, defined as$SO = \sum_{uv \in \mathbf E(G)}\sqrt{d_u^2+d_v^2}$, where$d_u$is the degree of the vertex$u$, and$\mathbf E(G)$is the edge set of$G$. Bounds on$SO$are established in terms of graph energy, size of minimum vertex cover, matching number, and induced matching number. • Pareto-efficient strategies in 3-person games played with staircase-function strategies • Abstract: A tractable method of solving 3-person games in which players&rsquo; pure strategies are staircase functions is suggested. The solution is meant to be Pareto-efficient. The method considers any 3-person staircase-function game as a succession of 3-person games in which strategies are constants. For a finite staircase-function game, each constant-strategy game is a trimatrix game whose size is likely to be relatively small to solve it in a reasonable time. It is proved that any staircase-function game has a single Pareto-efficient situation if every constant-strategy game has a single Pareto-efficient situation, and vice versa. Besides, it is proved that, whichever the staircase-function game continuity is, any Pareto-efficient situation of staircase function-strategies is a stack of successive Pareto-efficient situations in the constant-strategy games. If a staircase-function game has two or more Pareto-efficient situations, the best efficient situation is one which is the farthest from the triple of the most unprofitable payoffs. In terms of 0-1-standardization, the best efficient situation is the farthest from the triple of zero payoffs. • Further study on "an extended shortest path problem A data envelopment analysis approach" • Abstract: Amirteimoori proposed an approach based on data envelopment analysis (DEA) for &nbsp;multi-objective path problems on networks whose arcs contain multiple positive and negative attributes [A. Amirteimoori, An extended shortest path problem: A data envelopment analysis approach, Applied Mathematics Letters 25 (2012) 1839-1843]. The approach is to define a relative efficiency for each arcs using DEA models, and then to solve a longest path problem for obtaining a path with maximum efficiency. In this note, we focus on two drawbacks of the approach and illustrate them using examples. Then, we propose remedies to eliminate them. • Improved bounds for Kirchhoff index of graphs • Abstract: Let$G$be a simple connected graph with n vertices. The Kirchhoff index of$G$is defined as$Kf (G) = n\sum_{i=1}^{n-1}1/&mu;_i$, where$\mu_1\ge \mu_2\ge \dots\ge \mu_{n-1}&gt;\mu_n=0$are the Laplacian eigenvalues of$G$. Some bounds on$Kf (G)$in terms of graph parameters such as the number of vertices, the number of edges, first Zagreb index, forgotten topological index, etc., are presented. These bounds improve some previously known bounds in the literature. • Signed Bicyclic Graphs with Minimal Index • Abstract: The index &lambda;1(&Gamma;) of a signed graph &Gamma; = (G, &sigma;) is just the largest eigenvalue of its adjacency matrix. For any n ô°&ge; 4 we identify the signed graphs achieving the minimum index in the class of signed bicyclic graphs with n vertices. Apart from the n = 4 case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. A comparison of the algebraic connectivities of several T-shape and H-shape trees is performed through a signless Laplacian variant of the Jacobs-Trevisan algorithm, usually employed to count the adjacency eigenvalues lying in a given interval. As a by-product, all signed bicyclic graphs containing a theta-graph and whose index is less than 2 are detected. • Sombor index of some graph transformations • Abstract: The Sombor index of the graph G is a recently introduced degree based topological index. It is defined as SO = sum_{uv in E(G)} sqrt{d(u)^2+d(v)^2}, where d(u) isthe degree of the vertex u and E(G) is the edge set of G.In this paper we calculate SO of some graph transformations. • Covering total double Roman domination in graphs • Abstract: For a graph$G$with no isolated vertex, a covering total double Roman dominating function ($CTDRD$function)$f$of$G$is a total double Roman dominating function ($TDRD$function) of$G$for which the set$\{v\in V(G) f(v)\ne 0\}$is a vertex cover set.The covering total double Roman domination number$\gamma_{ctdR}(G)$equals the minimum weight of an$CTDRD$function on$G$. An$CTDRD$function on$G$with weight$\gamma_{ctdR} (G)$is called a$\gamma_{ctdR} (G)$-function. In this paper, the graphs$G$with small$\gamma_{ctdR} (G)$are characterised. We showthat the decision problem associated with$CTDRD$is$NP$-complete even when restricted to planer graphswith maximum degree at most four. We then show that for every graph$G$without isolated vertices,$\gamma_{oitR}(G)&lt;\gamma_{ctdR}(G)&lt; 2\gamma_{oitR}(G)$and for every tree$T$,$2\beta(T)+1\leq \gamma_{ctdR}(T)\leq4\beta(T)$, where$\gamma_{oitR}(G)$and$\beta(T)$are the outer independent total Roman domination number of$G$, and the minimum vertex cover number of$T$respectively. Moreover we investigate the$\gamma_{ctdR}$of corona of two graphs. • Efficient algorithms for independent Roman domination on some classes of graphs • Abstract: Let$G=(V,E)$be a given graph of order$n $. A function$f : V \to \{0,1, 2\}$is an independent Roman dominating function (IRDF) on$G$if for every vertex$v\in V$with$f(v)=0$there is a vertex$u$adjacent to$v$with$f(u)=2$and$\{v\in V:f(v)&gt; 0\}$is an independent set. The weight of an IRDF$f$on$G $is the value$f(V)=\sum_{v\in V}f(v)$. The minimum weight of an IRDF among all IRDFs on$G$is called the independent Roman domination number of~$G$. In this paper, we give algorithms for computing the independent Roman domination number of$G$in$O( V )$time when$G=(V,E)$is a tree, unicyclic graph or proper interval graph. • On the Zagreb indices of graphs with given Roman domination number • Abstract: Let$G$be a graph with vertex set$V(G)$and edge set$E(G)$. The two Zagreb indices$M_1=\sum_{v\in V(G)} d^2_G(v)$and$M_2=\sum_{uv\in E(G)} d_G(u)d_G(v)$are vertex degree based graph invariants that have been introduced in the 1970s and extensively studied ever since. {In this paper, we first give a lower bound on the first Zagreb index of trees with given Roman domination number and we characterize all extremal trees. Then we present upper bound for Zagreb indices of unicyclic and bicyclic graphs with given Roman domination number. • Bounds for fuzzy Zagreb Estrada index • Abstract: Let$G(V,\sigma ,\mu )$be a fuzzy graph of order$n$, where$\sigma(u)$is the vertex membership,$\mu(u,v)$is membership value of an edge and$\mu (u)$is the strength of vertex. The first fuzzy Zagreb index is the sum$\sigma ({{u}_{i}})\mu ({{u}_{i}})+\sigma ({{u}_{j}})\mu ({{u}_{j}})$where${{{u}_{i}}{{u}_{j}}\in {{\mu }}}$and the corresponding fuzzy Zagreb matrix is the square matrix of order$n$whose$(i,j)^{th}$entry whenever$i\neq j$, is$\sigma ({{u}_{i}})\mu ({{u}_{i}})+\sigma ({{u}_{j}})\mu ({{u}_{j}})$and zero otherwise. In this paper, we introduce the Zagreb Estrada index of fuzzy graphs and establish some bounds for it. • Unicyclic graphs with maximum Randić indices • Abstract: The Randi\'c index$R(G)$of a graph$G$is the sum of the weights$(d_u d_v)^{-\frac{1}{2}}$of all edges$uv$in$G$, where$d_u$denotes the degree of vertex$u$. Du and Zhou [On Randi\'c indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760--2770] determined the$n$-vertex unicyclic graphs with the third for$n\ge 5$, the fourth for$n\ge 7$and the fifth for$n\ge 8$maximum Randi\'c indices. Recently, Li et al. [The Randi{' c} indices of trees, unicyclic graphs and bicyclic graphs, Ars Combin. 127 (2016), 409--419] obtained the$n$-vertex unicyclic graphs with the sixth and the seventh for$n\ge 9$and the eighth for$n\ge 10$maximum Randi\'c indices. In this paper, we characterize the$n$-vertex unicyclic graphs with the ninth, the tenth, the eleventh, the twelfth and the thirteenth maximum Randi\'c values. • On perfectness of annihilating-ideal graph of Zn • Abstract: The annihilating-ideal graph of a commutative ring$R$with unity is defined as the graph$AG(R)$whose vertex set is the set of all non-zero ideals with non-zero annihilators and two distinct vertices$I$and$J$are adjacent if and only if$IJ = 0$. Nikandish et.al. proved that$AG(\mathbb{Z}_n)$is weakly perfect. In this short paper, we characterize$n$for which$AG(\mathbb{Z}_n)$is perfect. • Remarks on the restrained Italian domination number in graphs • Abstract: Let$G$be a graph with vertex set$V(G)$.An Italian dominating function (IDF) is a function$f:V(G)\longrightarrow \{0,1,2\}$having the property that that$f(N(u))\geq 2$for every vertex$u\in V(G)$with$f(u)=0$,where$N(u)$is the neighborhood of$u$. If$f$is an IDF on$G$, then let$V_0=\{v\in V(G): f(v)=0\}$. A restrained Italian dominating function (RIDF)is an Italian dominating function$f$having the property that the subgraph induced by$V_0$does not have an isolated vertex.The weight of an RIDF$f$is the sum$\sum_{v\in V(G)}f(v)$, and the minimum weight of an RIDF on a graph$G$is the restrained Italian domination number.We present sharp bounds for the restrained Italian domination number, and we determine the restrained Italian domination number for some families of graphs. • The stress of a graph • Abstract: Stress is an important centrality measure of graphs applicableto the study of social and biological networks. We study the stress of paths, cycles, fans andwheels. We determine the stress of a cut vertex of a graph G, when G has at most two cutvertices. We have also identified the graphs with minimum stress and maximum stress in thefamily of all trees of order$n$and in the family of all complete bipartite graphs of order n. • Some new bounds on the modified first Zagreb index • Abstract: Let$G$be a graph containing no isolated vertices. For the graph$G$, its modified first Zagreb index is defined as the sum of reciprocals of squares of vertex degrees of$G$. This article provides some new bounds on the modified first Zagreb index of$G$in terms of some other well-known graph invariants of$G$. From the obtained bounds, several known results follow directly. • Vertex-Edge Roman Domination in Graphs: Complexity and Algorithms • Abstract: For a simple, undirected graph$G(V,E)$, a function$h : V(G) \rightarrow \lbrace 0, 1, 2\rbrace$such that each edge$ (u,v)$of$G$is either incident with a vertex with weight at least one or there exists a vertex$w$such that either$(u,w) \in E(G)$or&nbsp;$(v,w) \in E(G)$and$h(w) = 2$, is called a vertex-edge Roman dominating function (ve-RDF) of$G$. For a graph$G$, the smallest possible weight of a ve-RDF of$G$which is denoted by$\gamma_{veR}(G)$, is known as the \textit{vertex-edge Roman domination number} of$G$. The problem of determining&nbsp;$\gamma_{veR}(G)$of a graph$G$is called minimum vertex-edge Roman domination problem (MVERDP). In this article, we show that the problem of deciding if$G$has a ve-RDF of weight at most$l$for star convex bipartite graphs, comb convex bipartite graphs, chordal graphs and planar graphs is NP-complete. On the positive side, we show that MVERDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. On the approximation point of view, a 2-approximation algorithm for MVERDP is presented. It is also shown that vertex cover and vertex-edge Roman domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming&nbsp; formulation for MVERDP is presented. • Signed total Italian k-domatic number of a graph • Abstract: Let$k\ge 1$be an integer, and let$G$be a finite and simple graph with vertex set$V(G)$.A&nbsp; signed total Italian$k$-dominating function on a graph$G$isa function$f:V(G)\longrightarrow \{-1, 1, 2\}$such that$\sum_{u\in N(v)}f(u)\ge k$for every$v\in V(G)$, where$N(v)$is the neighborhood of$v$, and each vertex$u$with$f(u)=-1$is adjacentto a vertex$v$with$f(v)=2$or to two vertices$w$and$z$with$f(w)=f(z)=1$.A set$\{f_1,f_2,\ldots,f_d\}$of distinct signed total Italian$k$-dominatingfunctions on$G$with the property that$\sum_{i=1}^df_i(v)\le k$for each$v\in V(G)$, is called a&nbsp;signed total Italian$k$-dominating family (of functions) on$G$. The maximum number of functionsin a signed total Italian$k$-dominating family on$G$is the&nbsp; signed total Italian k-domatic number of$G$,&nbsp;denoted by$d_{stI}^k(G)$. In this paper we initiate the study of signed total Italian k-domatic numbers&nbsp;in graphs, and we present sharp bounds for$d_{stI}^k(G)$. In addition, we determine the signed total Italian&nbsp;k-domatic number of some graphs. • Restrained double Italian domination in graphs • Abstract: Let$G$be a graph with vertex set$V(G)$.A double Italian dominating function (DIDF) is a function$f:V(G)\longrightarrow \{0,1,2,3\}$having the property that$f(N[u])\geq 3$for every vertex$u\in V(G)$with$f(u)\in \{0,1\}$,where$N[u]$is the closed neighborhood of$u$. If$f$is a DIDF on$G$, then let$V_0=\{v\in V(G): f(v)=0\}$. A restrained double Italian dominating function (RDIDF)is a double Italian dominating function$f$having the property that the subgraph induced by$V_0$does not have an isolated vertex.The weight of an RDIDF$f$is the sum$\sum_{v\in V(G)}f(v)$, and the minimum weight of an RDIDF on a graph$G$is the restrained double Italian domination number.We present bounds and Nordhaus-Gaddum type results for the restrained double Italian domination number. In addition, we determine therestrained double Italian domination number for some families of graphs. • On the distance spectra of product of signed graphs • Abstract: In this article, we study the distance matrix of the product of signed graphs such as the Cartesian product and the lexicographic product in terms of the signed distance matrices of the factor graphs. Also, we discuss the signed distance spectra of some special classes of product of signed graphs. • Some families of$\alpha$-labeled subgraphs of the integral grid • Abstract: In this work we study the most restrictive variety of graceful labelings, that is, we study the existence of an$\alpha$-labeling for some families of graphs that can be embedded in the integral grid. Among the categories of graphs considered here we have a subfamily of 2-link fences, a subfamily of column-convex polyominoes, and a subfamily of irregular cyclic-snakes. We prove that under some conditions, the a-labelings of these graphs can be transformed into harmonious labelings. We also present a closed formula for the number of 2-link fences examined here. • On Randić spectrum of zero divisor graphs of commutative ring$\mathbb{Z}_{n} $• Abstract: For a finite commutative ring$ \mathbb{Z}_{n} $with identity$ 1\neq 0 $, the zero divisor graph$ \Gamma(\mathbb{Z}_{n}) $is a simple connected graph having vertex set as the set of non-zero zero divisors, where two vertices$ x $and$ y $are adjacent if and only if$ xy=0 $. We find the Randi\'c spectrum of the zero divisor graphs$ \Gamma(\mathbb{Z}_{n}) $, for various values of$ n$and characterize$ n $for which$ \Gamma(\mathbb{Z}_{n}) \$ is Randi\'c integral.

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