Abstract: Let k ≥ 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 (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertices w and z with f(w)=f(z)=1. The weight of an STIkDF f is$omega(f)=sum_{vin V(G)}f(v)$. The signed total Italian k-domination number $gamma_{stI}^k(G)$ of G is the minimum weight of an STIkDF on G. In this paper we initiate the study of the signed total Italian k-dominationnumber of graphs, and we present different bounds on $gamma_{stI}^k(G)$. In addition, we determine thesigned total Italian k-domination number of some classes of graphs. Some of our results are extensions ofwell-known properties of the signed total Roman $k$-domination number $gamma_{stR}^k(G)$,introduced and investigated by Volkmann [9,12].

Abstract: ÞFor a given graph G, its Þ-energy is the sum of the absolute values of the eigenvalues of the Þ-matrix of G. In this article, we explore the Þ-energy of generalized Petersen graphs G(p,k) for various vertex partitions such as independent, domatic, total domatic and k-ply domatic partitions and partition containing a perfect matching in G(p,k). Further, we present a python program to obtain the Þ-energy of G(p,k) for the vertex partitions under consideration and examine the relation between them.

Abstract: Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π is called the distinct edge geodetic decomposition number of G and is denoted by π_dg1 (G), where g_1 (G) is the edge geodetic number of G. Some general properties satisfied by this concept are studied. Connected graphs which are edge geodetic decomposable are characterized. Connected distinct edge geodetic decomposable graphs of order p with π_dg1 (G)= p-2 are characterised.

Abstract: Let k≥ 1 be an integer. A weak signed Roman k-dominating function on a graph G isa function f:V (G)→ {-1, 1, 2} such that ΣuεN[v] f(u)≥ k for everyvε V(G), where N[v] is the closed neighborhood of v.A set {f1,f2, ... ,fd} of distinct weak signed Roman k-dominatingfunctions on G with the property that Σ1≤i≤d fi(v)≤ k for each vε V(G), is called a weak signed Roman k-dominating family (of functions) on G. The maximum number of functionsin a weak signed Roman k-dominating family on G is the weak signed Roman k-domatic number} of Gdenoted by dwsR k(G). In this paper we initiate the study of the weak signed Roman $k$-domatic numberin graphs, and we present sharp bounds for dwsR k(G). In addition, we determine the weak signed Roman k-domatic number of some graphs.

Abstract: A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_{vin V(G)}f(v)$ among all total Roman dominating functions $f$ on $G$.It is known that $gamma_{tR}(G)geq gamma_{t2}(G)+gamma(G)$ for any graph $G$ with neither isolated vertex nor components isomorphic to $K_2$, where $gamma_{t2}(G)$ and $gamma(G)$ represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.

Abstract: In this paper, we present a second-order corrector infeasibleinterior-point method for linear optimization in a largeneighborhood of the central path. The innovation of our method is tocalculate the predictor directions using a specific kernel functioninstead of the logarithmic barrier function. We decompose thepredictor direction induced by the kernel function to two orthogonaldirections of the corresponding to the negative and positivecomponent of the right-hand side vector of the centering equation.The method then considers the new point as a linear combination ofthese directions along with a second-order corrector direction. Theconvergence analysis of the proposed method is investigated and itis proved that the complexity bound is Ο(n5/4 log ε-1).

Abstract: The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

Abstract: A signed graph is a graph in which each edge has a positive or negative sign. In this article, we define n^th power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an n^th power of a signed graph to be unique. Also, we characterize balanced power signed graphs.

Abstract: If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G)=α D(G)+(1-α) A(G), where 0≤ α ≤ 1. The Aα (or generalized) spectral radius λ(Aα(G)) (or simply λα) is the largest eigenvalue of Aα(G). In this paper, we show that λα ≤αΔ+(1-α)(2m(1-1/ω))1/2, where m, Δ and ω=ω(G) are respectively the size, the largest degree and the clique number of $G$. Further, if G has order n, then we show that
2λα ≤ max1≤i≤n [αdi + √α2 di ^2 +4mi(1-α)[α+(1-α)mj]
where di and mi are respectively the degree and the average 2-degree of the vertex vi.

Abstract: In this paper, we initiate the study of total outer-convex domination as a new variant of graph domination and we show the close relationship that exists between this novel parameter and other domination parameters of a graph such as total domination, convex domination, and outer-convex domination. Furthermore, we obtain general bounds of total outer-convex domination number and, for some particular families of graphs, we obtain closed formulas.

Abstract: To extract some more information from the constructions of matroids that arise from new operations, computing the Tutte polynomial, plays an important role. In this paper, we consider applying three operations of splitting, element splitting and splitting off to a binary matroid and then introduce the Tutte polynomial of resulting matroids by these operations in terms of that of original matroids.

Abstract: A subset D of vertices of a graph G is a dominating set if for each u ∈ V (G) \ D, u is adjacent to somevertex v ∈ D. The domination number, γ(G) ofG, is the minimum cardinality of a dominating set of G. A setD ⊆ V (G) is a total dominating set if for eachu ∈ V (G), u is adjacent to some vertex v ∈ D. Thetotal domination number, γt (G) of G, is theminimum cardinality of a total dominating set of G. For an eveninteger $nge 2$ and $1\le Delta \le lfloorlog_2nrfloor$, aKnodel graph $W_{Delta,n}$ is a $Delta$-regularbipartite graph of even order n, with vertices (i,j), for$i=1,2$ and $0le jle n/2-1$, where for every $j$, $0le jlen/2-1$, there is an edge between vertex $(1, j)$ and every vertex$(2,(j+2^k-1)$ mod (n/2)), for $k=0,1,cdots,Delta-1$. In thispaper, we determine the total domination number in $3$-regularKnodel graphs $W_{3,n}$.

Abstract: Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

Abstract: The variable sum exdeg index of a graph G is defined as $SEI_a(G)=sum_{uin V(G)}d_G(u)a^{d_G(u)}$, where $aneq 1$ is a positive real number, du(u) is the degree of a vertex u ∈ V (G). In this paper, we characterize the graphs with the extremum variable sum exdeg index among all the graphs having a fixed number of vertices and cut edges, for every a>1.

Abstract: Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signless Laplacian matrix of graph $G$. Theeigenvalues of matrix $mathcal{L}^{+}(G)$, $2=gamma _{1}^{+}geq gamma_{2}^{+}geq cdots geq gamma _{n}^{+}geq 0$, are normalized signlessLaplacian eigenvalues of $G$. In this paper some bounds for the sum $%K^{+}(G)=sum_{i=1}^nfrac{1}{gamma _{i}^{+}}$ are considered.

Abstract: ‎A Roman dominating function (RDF) on a graph G=(V,E) is a function f : V → {0, 1, 2} such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over all vertices‎, ‎and the outer‎‎independent Roman domination number ΥoiR (G) is the minimum weight‎‎of an OIRDF on $G$‎. ‎In this paper‎, ‎we show that if T is a tree of order n ≥ 3 with s(T) support vertices‎, ‎then $gamma _{oiR}(T)leq min‎ {‎frac{5n}{6},frac{3n+s(T)}{4}}.$ Moreover‎, ‎we characterize the tress‎‎attaining each bound‎.

Abstract: A set S of vertices in a graph G=(V,E) is a dominating set ofG if every vertex of V-S is adjacent to some vertex of S.For an integer k≥1, a set S of vertices is a k-step dominating set if any vertex of $G$ is at distance k from somevertex of S. In this paper, using membership values of vertices and edges in fuzzy graphs, we introduce the concepts of strength of strongestdominating set as well as strength of strongest$k$-step dominating set in fuzzy graphs. We determine various bounds forthese parameters in fuzzy graphs. We also determine the strengthof strongest dominating set in some families of fuzzy graphsincluding complete fuzzy graphs and complete bipartite fuzzygraphs.

Abstract: The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) is a complete graph. In this paper, we find the line completion number of grid graph Pn × Pm for various cases of n and m.

Abstract: ‎Let G be a graph‎. ‎A 2-rainbow dominating function (or‎ 2-RDF) of G is a function f from V(G)‎ ‎to the set of all subsets of the set {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled‎, wher NG(v) is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎$omega (f):=sum _{vin V(G)} f(v) $‎. ‎The 2-rainbow‎‎domination number of G‎, ‎denoted by Υr2 (G)‎, ‎is the‎‎minimum weight of a 2-RDF of G‎. ‎A 2-RDF f is called an outer independent 2-rainbow dominating function ‎(or OI2-RDF) of G if‎‎the set of all v ∈ V (G) with f(v) = ∅ is an‎ ‎independent set‎. ‎The outer independent 2-rainbow domination number Υoir2 (G) is‎‎the minimum weight of an OI2-RDF of G‎. ‎In this paper‎, ‎we obtain the‎‎outer independent 2-rainbow domination number of Pm□Pn‎ ‎and‎ Pm□Cn‎. ‎Also we determine the value of Υoir2 (Cm2Cn) when m or n is even‎.

Abstract: The energy of a graph G, denoted by Ε(G), is defined as the sum of the absolute values of all eigenvalues of G. In this paper, lower and upper bounds for energy in some of the graphs are established, in terms of graph invariants such as the number of vertices, the number of edges, and the number of closed walks.

Abstract: In this short note, we disprove the conjecture of Jafari Rad and Volkmann that every γ-vertex critical graph is γR-vertex critical, where γ(G) and γR(G) stand for the domination number and the Roman domination number of a graph G, respectively.

Abstract: For a simple, undirected, connected graph G=(V,E), a function f : V(G) →{0, 1, 2} which satisfies the following conditions is called a quasi-total Roman dominating function (QTRDF) of G with weight f(V(G))=ΣvΕV(G) f(v).C1). Every vertex uεV for which f(u) = 0 must be adjacent to at least one vertex v with f(v) = 2, and C2). Every vertex uεV for which f(u) = 2 must be adjacent to at least one vertex v with f(v)≥1. For a graph G, the smallest possible weight of a QTRDF of G denoted γqtR(G) is known as the quasi-total Roman domination number of G. The problem of determining γqtR(G) of a graph G is called minimum quasi-total Roman domination problem (MQTRDP). In this paper, we show that the problem of determining whether G has a QTRDF of weight at most l is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. On the positive side, we show that MQTRDP for threshold graphs, chain graphs and bounded treewidth graphs is linear time solvable. Finally, an integer linear programming formulation for MQTRDP is presented.

Abstract: A subset S of vertices in a graph G = (V;E) is 2-independent if every vertexof S has at most one neighbor in S: The 2-independence number is the maximumcardinality of a 2-independent set of G: In this paper, we initiate the study of the2-independence subdivision number sdβ2(G) defined as the minimum numberof edges that must be subdivided (each edge in G can be subdivided at mostonce) in order to increase the 2-independence number. We first show that forevery connected graph G of order at least three, 1≤sdβ2(G)≤2; and we give anecessary and sufficient condition for graphs G attaining each bound. Moreover,restricted to the class of trees, we provide a constructive characterization of alltrees T with sdβ2(T)= 2; and we show that such a characterization suggestsan algorithm that determines whether a tree T has sdβ2(T)= 2 or sdβ2(T) = 1in polynomial time.

Abstract: The chromatic number, Χ(G) of a graph G is the minimum number of colours used in a proper colouring of G. In improper colouring, an edge uv is bad if the colours assigned to the end vertices of the edge is the same. Now, if the available colours are less than that of the chromatic number of graph G, then colouring the graph with the available colours leads to bad edges in G. The number of bad edges resulting from a δ(k)-colouring of G is denoted by bk(G). In this paper, we use the concept of δ(k)-colouring and determine the number of bad edges in the Cartesian product of some graphs.