Closed Neighborhood Degree Sum-Based Topological Descriptors of Graphene Structures

Topological descriptors defined on chemical structures enable understanding the properties and activities of chemical molecules. In this paper, we compute closed neighborhood degree sum-based indices for four different Graphene structures. The cardinality of closed neighborhood degree-based edge partitions for four different Graphene structures is used to compute the closed neighborhood degree sum-based indices.

Solutions to four open problems on quorum colorings of graphs

A partition $\pi=\{V_{1},V_{2},...,V_{k}\}$ of the vertex set $V$ of a graph $G$ into $k$ color classes $V_{i},$ with $1\leq i\leq k$ is called a quorum coloring if for every vertex $v\in V,$ at least half of the vertices in the closed neighborhood $N[v]$ of $v$ have the same color as $v$. The maximum cardinality of a quorum coloring of $G$ is called the quorum coloring number of $G$ and is denoted $\psi_{q}(G).$ In this paper, we give answers to four open problems stated in 2013 by Hedetniemi, Hedetniemi, Laskar and Mulder. In particular, we show that there is no good characterization of the graphs $G$ with $\psi_{q}(G)=1$ nor for those with $\psi_{q} (G)>1$ unless $\mathcal{P}\neq\mathcal{NP}\cap co-\mathcal{NP}.$ We also construct several new infinite families of such graphs, one of which the diameter $diam(G)$ of $G$ is not bounded.

LOCALLY-BALANCED $k$-PARTITIONS OF GRAPHS

In this paper we generalize locally-balanced $2$-partitions of graphs and introduce a new notion, the locally-balanced $k$-partitions of graphs, defined as follows: a $k$-partition of a graph $G$ is a surjection $f:V(G)\rightarrow \{0,1,\ldots,k-1\}$. A $k$-partition ($k\geq 2$) $f$ of a graph $G$ is a locally-balanced with an open neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=i\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=j\}\vert \right\vert\leq 1.$$ A $k$-partition ($k\geq 2$) $f^{\prime}$ of a graph $G$ is a locally-balanced with a closed neighborhood, if for every $v\in V(G)$ and any $0\leq i<j\leq k-1$ $$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=i\}\vert - \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=j\}\vert \right\vert\leq 1.$$ The minimum number $k$ ($k\geq 2$), for which a graph $G$ has a locally-balanced $k$-partition with an open (a closed) neighborhood, is called an $lb$-open ($lb$-closed) chromatic number of $G$ and denoted by $\chi_{(lb)}(G)$ ($\chi_{[lb]}(G)$). In this paper we determine or bound the $lb$-open and $lb$-closed chromatic numbers of several families of graphs. We also consider the connections of $lb$-open and $lb$-closed chromatic numbers of graphs with other chromatic numbers such as injective and $2$-distance chromatic numbers.

Approximation algorithms for maximum weight k-coverings of graphs by packings

Let [Formula: see text] be a family of graphs and let [Formula: see text] be a set of connected graphs, each with at most [Formula: see text] vertices, [Formula: see text] fixed. A [Formula: see text]-packing of a graph GA is a vertex induced subgraph of GA with every connected component isomorphic to a member of [Formula: see text]. A maximum weight [Formula: see text]-covering of a graph GA by [Formula: see text]-packings, is a maximum weight subgraph of GA exactly covered by [Formula: see text] vertex disjoint [Formula: see text]-packings. For a graph [Formula: see text] let [Formula: see text](GA) be a graph, every vertex [Formula: see text] of which corresponds to a vertex subgraph [Formula: see text] of GA isomorphic to a member of [Formula: see text], two vertices [Formula: see text] of [Formula: see text](GA) being adjacent if and only if [Formula: see text] and [Formula: see text] have common vertices or interconnecting edges. The closed neighborhoods containment graph [Formula: see text] of a graph [Formula: see text], is the graph with vertex set [Formula: see text] and edges directed from vertices [Formula: see text] to [Formula: see text] if and only if they are adjacent in GA and the closed neighborhood of [Formula: see text] is contained in the closed neighborhood of [Formula: see text]. A graph [Formula: see text] is a [Formula: see text] reduced graph if it can be obtained from a graph [Formula: see text] by deleting the edges of a transitive subgraph [Formula: see text] of CNCG(GA). We describe 1.582-approximation algorithms for maximum weight [Formula: see text]-coverings by [Formula: see text]-packings of [Formula: see text] and [Formula: see text] reduced graphs when [Formula: see text] is vertex hereditary, has an algorithm for maximum weight independent set and [Formula: see text]. These algorithms can be applied to families of interval filament, subtree filament, weakly chordal, AT-free and circle graphs, to find 1.582 approximate maximum weight [Formula: see text]-coverings by vertex disjoint induced matchings, dissociation sets, forests whose subtrees have at most [Formula: see text] vertices, etc.

A lower bound on the global powerful alliance number in trees

For a graph $G=(V,E)$, a set $D\subseteq V$ is a dominating set if every vertex in $V-D$ is either in $D$ or has a neighbor in $D$. A dominating set $D$ is a global offensive alliance (resp. a global defensive alliance) if for each vertex $v$ in $V-D$ (resp. $v$ in $D$) at least half the vertices from the closed neighborhood of $v$ are in $D$. A global powerful alliance is both global defensive and global offensive. The global powerful alliance number $\gamma_{pa}(G)$ is the minimum cardinality of a global powerful alliance of $G$. We show that if $T$ is a tree of order $n$ with $l$ leaves and $s$ support vertices, then $\gamma_{pa}(T)\geq\frac{3n-2l-s+2}{5}$. Moreover, we provide a constructive characterization of all extremal trees attaining this bound.

Remarks on the outer-independent double Italian domination number

Let \(G\) be a graph with vertex set \(V(G)\). If \(u\in V(G)\), then \(N[u]\) is the closed neighborhood of \(u\). An outer-independent double Italian dominating function (OIDIDF) on a graph \(G\) is a function \(f:V(G)\longrightarrow \{0,1,2,3\}\) such that if \(f(v)\in\{0,1\}\) for a vertex \(v\in V(G)\), then \(\sum_{x\in N[v]}f(x)\ge 3\), and the set \(\{u\in V(G):f(u)=0\}\) is independent. The weight of an OIDIDF \(f\) is the sum \(\sum_{v\in V(G)}f(v)\). The outer-independent double Italian domination number \(\gamma_{oidI}(G)\) equals the minimum weight of an OIDIDF on \(G\). In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs.

Majority Roman domination in graphs

A Majority Roman Dominating Function (MRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that (i) the sum of its function values over at least half the closed neighborhood is at least one and (ii) every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a MRDF is the sum of its function values over all vertices. The Majority Roman Domination Number of a graph [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text]. In this paper, we initiate the study of Majority Roman Domination in Graphs.

Certain types of m-polar interval-valued fuzzy graph

In this paper, an unprecedented kind of fuzzy graph designated as m-polar interval valued fuzzy graph (m-PIVFG) is defined. Complement of the m-PIVFG open and closed neighborhood degrees of m-PIVFG are discussed. The other algebraic properties such as density, regularity, irregularity of the m-PIVFG are investigated. Moreover, some basic results on regularity and irregularity of m-PIVFG are proved. Free nodes and busy nodes of m-PIVFG is explored with some basic theorems and examples. Lastly, an application of m-PIVFG is described.

Double Domination Number of Some Families of Graph

In a graph G = (V, E) each vertex is said to dominate every vertex in its closed neighborhood. In a graph G, if S is a subset of V then S is a double dominating set of G if every vertex in V is dominated by at least two vertices in S. The smallest cardinality of a double dominating set is called the double domination number γx2 (G). [4]. In this paper, we computed some relations between double domination number, domination number, number of vertices (n) and maximum degree (∆) of Helm graph, Friendship graph, Ladder graph, Circular Ladder graph, Barbell graph, Gear graph and Firecracker graph.