Related papers: Optimal Graphs for Independence and $k$-Independen…

A complete subgraph of any simple graph $G$ on $k$ vertices is called a $k$-\emph{clique} of $G$. In this paper, we first introduce the concept of the value of a $k$-clique ($k>1$) as an extension of the idea of the degree of a given…

An independent broadcast on a graph $G$ is a function $f: V \longrightarrow \{0,\ldots,{\rm diam}(G)\}$ such that $(i)$ $f(v)\leq e(v)$ for every vertex $v\in V(G)$, where $\operatorname{diam}(G)$ denotes the diameter of $G$ and $e(v)$ the…

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent…

Erd\H{o}s, Hajnal and Szemer\'{e}di proved that any subset $G$ of vertices of a shift graph $\text{Sh}_{n}^{k}$ has the property that the independence number of the subgraph induced by $G$ satisfies $\alpha(\text{Sh}_{n}^{k}[G])\geq…

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number…

We study the minimal homogeneous generating sets of the Eulerian ideal associated with a simple graph and its maximal generating degree. We show that the Eulerian ideal is a lattice ideal and use this to give a characterization of binomials…

We present quantum complexity lower and upper bounds for independent set problems in graphs. In particular, we give quantum algorithms for computing a maximal and a maximum independent set in a graph. We present applications of these…

An independent set in a graph G is a set of vertices no two of which are joined by an edge. A vertex-weighted graph associates a weight with every vertex in the graph. A vertex-weighted graph G is called a unique independence…

The Cayley sum graph $\Gamma_S$ of a set $S \subseteq \mathbb{Z}_n$ is defined on the vertex set $\mathbb{Z}_n$, with an edge between distinct $x, y \in \mathbb{Z}_n$ if $x + y \in S$. Campos, Dahia, and Marciano have recently shown that if…

For every graph $X$, we consider the class of all connected $\{K_{1,3}, X\}$-free graphs which are distinct from an odd cycle and have independence number at least $4$, and we show that all graphs in the class are perfect if and only if $X$…

The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double…

An independent coalition in a graph $G$ consists of two disjoint sets of vertices $V_1$ and $V_2$ neither of which is an independent dominating set but whose union $V_1 \cup V_2$ is an independent dominating set. An independent coalition…

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…

Write ${\cal I}(G)$ for the set of independent sets of a graph $G$ and $i(G)$ for $|{\cal I}(G)|$. It has been conjectured (by Alon and Kahn) that for an $N$-vertex, $d$-regular graph $G$, $$ i(G) \leq \left(2^{d+1}-1\right)^{N/2d}. $$ If…

A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{independence number} $\alpha(G)$ is the cardinality of a maximum independent set, while $\mu(G)$ is the…

Let $G$ be a group. The power graph of $G$ is a graph with vertex set $G$ in which two distinct elements $x,y$ are adjacent if one of them is a power of the other. We characterize all groups whose power graphs have finite independence…

We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex…

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the…

A graph is equimatchable if each of its matchings is a subset of a maximum matching. It is known that any 2-connected equimatchable graph is either bipartite, or factor-critical, and that these two classes are disjoint. This paper provides…

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…