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.

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.

Abstract: The $2S3$ transformation, which was first described for positive integers, has been defined for dyadic rational numbers in the open interval $(0,1)$ in this study. The set of dyadic rational numbers is a Prüfer 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.

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.

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 $RDF$ $f$ 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)].

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.

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.

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 $c(G-e)< c(G)$ or $c(G-e)>c(G)$. In this article, we study the edge criticality of generalized Petersen graphs and Paley graphs.

Abstract: In this paper, we have punctured unit $\mathbb{Z}_q$-Simplex code and constructed a new code called unit $\mathbb{Z}_q$-Simplex code of type $\alpha$. In particular, we find the parameters of 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 $q.$ For $q$ is a prime power and rank $k=3,$ we have given the weight distribution of unit $\mathbb{Z}_q$-Simplex codes of type $\alpha$. Also, we have introduced some new code from $\mathbb{Z}_q$-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 weight distribution of zero divisor $\mathbb{Z}_q$-Simplex code for rank $k=3$ and $q$ is a prime power.

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.

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.

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’s domination number is the cardinality of a minimum total liar’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.

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.

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 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 for any $\delta > 0$ unless $P = NP$. Next, we show that MTR2DP is APX-hard for graphs with $ \Delta=4$. Finally, we show that the domination and TR2DF problems are not equivalent in computational complexity aspects.

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.

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 $F(G)>88$ and with $HM(G)>158$. Along with the result, we show that there exist no trees with $F(G) < 90$ and $HM(G) < 160$ with some exceptional even positive integers and hence characterize the forgotten Zagreb index and the hyper Zagreb index for trees.

Abstract: For a connected graph $G$ of order at least two, a 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 $m_{oc}(G)$. The number of extreme vertices in $G$ is its \textit{extreme order} $ex(G)$. A graph $G$ is said to be an \textit{extreme outer connected monophonic graph} if $m_{oc}(G)$ = $ex(G)$. 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. 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 for positive integers $r,d$ and $k \geq 2$ with $r < d$, there exists an extreme outer connected monophonic graph $G$ with monophonic radius $r$, monophonic diameter $d$ and outer connected monophonic number $k$.

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.

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.

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.

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. 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.

Abstract: ‎‎For an integer $k\geq 2$‎, ‎a Roman $k$-tuple dominating function‎, ‎(or just RkDF)‎, ‎in a graph $G$ is a function $f \colon V(G) \rightarrow \{0‎, ‎1‎, ‎2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least $k$ vertices $v$ for which $f(v) = 2$‎, ‎and every vertex $u$ for which $f(u) \neq 0$ is adjacent to at least $k-1$ vertices $v$ for which $f(v) = 2$‎. ‎The Roman $k$-tuple domination number of ‎$‎G‎$‎‎ ‎is the minimum weight of an RkDF in $G$. ‎In this note we settle two problems posed in [Roman $k$-tuple Domination in Graphs‎, ‎Iranian J‎. ‎Math‎. ‎Sci‎. ‎Inform‎. ‎15 (2020)‎, ‎101--115]‎.

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.

Abstract: A $k$-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 $k$.

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.

Abstract: A tractable method of solving 3-person games in which players’ 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.

Abstract: Amirteimoori proposed an approach based on data envelopment analysis (DEA) for 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.

Abstract: Let $G$ be a simple connected graph with n vertices. The Kirchhoff index of $G$ is defined as $Kf (G) = n\sum_{i=1}^{n-1}1/μ_i$, where $\mu_1\ge \mu_2\ge \dots\ge \mu_{n-1}>\mu_n=0$ 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.

Abstract: The index λ1(Γ) of a signed graph Γ = (G, σ) is just the largest eigenvalue of its adjacency matrix. For any n ô°≥ 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.

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.

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, $\gamma_{oitR}(G)<\gamma_{ctdR}(G)< 2\gamma_{oitR}(G)$ 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.

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 $\{v\in V:f(v)> 0\}$ 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.

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.

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.

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.

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.

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.

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.

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.

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 $(v,w) \in E(G)$ 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 $\gamma_{veR}(G)$ 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 formulation for MVERDP is presented.

Abstract: Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A 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 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 signed total Italian k-domatic number of $G$, denoted by $d_{stI}^k(G)$. In this paper we initiate the study of signed total Italian k-domatic numbers in graphs, and we present sharp bounds for $d_{stI}^k(G)$. In addition, we determine the signed total Italian k-domatic number of some 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.

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.

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.

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.