Related papers: On Weak Integer Additive Set-Indexers of Certain G…

Let $1 \leq k \leq n$ be a positive integer. A {\em nonnegative signed $k$-subdominating function} is a function $f:V(G) \rightarrow \{-1,1\}$ satisfying $\sum_{u\in N_G[v]}f(u) \geq 0$ for at least $k$ vertices $v$ of $G$. The value…

Graph product is a fundamental tool with rich applications in both graph theory and theoretical computer science. It is usually studied in the form $f(G*H)$ where $G$ and $H$ are graphs, * is a graph product and $f$ is a graph property. For…

Frank Harary introduced the concept of integral sum graph. A graph $G$ is an \emph{ integral sum graph} if its vertices can be labeled with distinct integers so that $e = uv$ is an edge of $G$ if and only if the sum of the labels on…

Menger's theorem is an important building block of numerous results in the study of graph structure. We consider a variant in terms of coarse geometry. We say that a set of graphs has the weak coarse Menger property if there exist functions…

A set $D$ of vertices in a graph $G$ is a dominating set if every vertex of $G$, which is not in $D$, has a neighbor in $D$. A set of vertices $D$ in $G$ is convex (respectively, isometric), if all vertices in all shortest paths…

The modified Albertson index, denoted by $A\!^*\!$, of a graph $G$ is defined as $A\!^*\!(G)=\sum_{uv\in E(G)} |(d_{u})^{2}- (d_{v})^{2}|$, where $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively, of $G$ and $E(G)$ is…

The regularity of an edge ideal of a finite simple graph $G$ is at least the induced matching number of $G$ and is at most the minimum matching number of $G$. If $G$ possesses a dominating inuduced matching, i.e., an induced matching which…

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th…

A subset $M$ of the edge set of a graph $G$ is an induced matching of $G$ if given any two $e_1,e_2 \in M$, none of the vertices on $e_1$ is adjacent to any of the vertices on $e_2$. Suppose that $MIM_G$, a positive integer, is the largest…

For a positive integer $t$ and a graph $G$, an additive $t$-spanner of $G$ is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus $t$. Minimum Additive $t$-Spanner Problem is to…

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$…

Let $\eta$ be a fixed positive integer. Let $S$ be a subset of $\mathbb{Z}$, $\star:S\times S\to \mathbb{Z}$ be a binary function, and $\zeta_{\eta}:\{\xi\in \mathbb{Z}:\gcd(\xi,\eta)=1\}\to \{0,1\}$ be a function. For a simple connected…

A dominating induced matching, also called an efficient edge domination, of a graph $G=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges is a subset $F \subseteq E$ of edges in the graph such that no two edges in $F$ share a common endpoint…

If $G(M)$ denotes the subgraph of a graph $G$ induced by the set of vertices that are covered by some matching $M$ in $G$, then $M$ is an induced or a uniquely restricted matching if $G(M)$ is $1$-regular or if $M$ is the unique perfect…

Given an undirected unweighted graph $G = (V, E)$ on $n$ vertices and $m$ edges, a subgraph $H\subseteq G$ is a spanner of $G$ with stretch function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, if for every pair $s, t$ of vertices in $V$,…

A dominating set $D_{f}\subseteq V(G)$ of vertices in a graph $G$ is called a \emph{dom-forcing set} if the sub-graph induced by $\langle D_{f} \rangle$ must form a zero forcing set. The minimum cardinality of such a set is known as the…

Transmission of a vertex v of a connected graph G is the sum of distances from v to all other vertices in G. Graph G is transmission irregular (TI) if no two of its vertices have the same transmission, and G is interval transmission…

An open packing in a graph $G$ is a set $S$ of vertices in $G$ such that no two vertices in $S$ have a common neighbor in $G$. The injective chromatic number $\chi_i(G)$ of $G$ is the smallest number of colors assigned to vertices of $G$…

Given a graph $F$, a graph $G$ is weakly $F$-saturated if all non-edges of $G$ can be added in some order so that each new edge introduces a copy of $F$. The weak saturation number $\operatorname{wsat}(n, F)$ is the minimum number of edges…

Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced…