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 NG(u) ∩ L ≥ 2 for any u ε V(G) and NG(u) ∩ NG(v) ∩ L ≥ 3 for any distinct vertices u,v ε V(G). The total liar’s domination number is the cardinality of a minimum total liar’s dominating set of $G$ and is denoted by γTLR(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 minimumnumber of edges that must be subdivided (where each edge in $G$ can besubdivided at most once) in order to increase the paired domination numberof $G$. In this note, we show that the problem of computing thepaired-domination subdivision number is NP-hard for bipartite graphs.

Abstract: For a simple, undirected, connected graph G, a function h : V → {0, 1, 2} 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 NG(v) with weight 2, or at least two vertices x, y in NG(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 ∑pεV 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(Δ - 0.5) + 1.5)-approximation algorithm for the MTR2DP and show that the same cannot have (1-δ) ln( V ) ratio approximation algorithm for any δ > 0 unless P=Np . Next, we show that MTR2DP is APX-hard for graphs with Δ=4. We also 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 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. {\bf 258} (2019), 235--241], Volkmann defined the \textit{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: 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.

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 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: 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: 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: 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.