Related papers: Properties of $\theta$-super positive graphs

Let $G$ be a group. The \emph{power graph} of $G$ is a graph with the vertex set $G$, having an edge between two elements whenever one is a power of the other. We characterize nilpotent groups whose power graphs have finite independence…

A subset $M$ of the edges of a graph $G$ is a matching if no two edges in $M$ are incident. A maximal matching is a matching that is not contained in a larger matching. A subset $S$ of vertices of a graph $G$ with no isolated vertices is a…

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if…

A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$…

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs,…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

Let $F_G(P)$ be a functional defined on the set of all the probability distributions on the vertex set of a graph $G$. We say that $G$ is \emph{symmetric with respect to $F_G(P)$} if the uniform distribution on $V(G)$ maximizes $F_G(P)$.…

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…

A graph $G$ is $F$-free if $G$ does not contain $F$ as a subgraph. Let $\mathcal{G}(m, F)$ denote the family of $F$-free graphs with $m$ edges and without isolated vertices. Let $S_{n,k}$ denote the graph obtained by joining every vertex of…

A square (0,1)-matrix X of order n > 0 is called fully indecomposable if there exists no integer k with 0 < k < n, such that X has a k by n-k zero submatrix. A stable set of a graph G is a subset of pairwise nonadjacent vertices. The…

One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…

A graph $G$ is well-covered if all maximal independent sets are of the same cardinality. Let $w:V(G) \longrightarrow\mathbb{R}$ be a weight function. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. An…

A set S is independent in a graph G if no two vertices from S are adjacent. By core(G) we mean the intersection of all maximum independent sets. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G)…

A graph $G$ is called matching covered if all of its edges are contained in some perfect matching of $G$. Furthermore, a cycle $C \subseteq G$ is called conformal if $G - V(C)$ has a perfect matching and $G$ itself is called cycle-conformal…

Given a graph $G$ with vertices $\{v_1,\ldots,v_n\}$, we define $\mathcal{S}(G)$ to be the set of symmetric matrices $A=[a_{i,j}]$ such that for $i\ne j$ we have $a_{i,j}\ne 0$ if and only if $v_iv_j\in E(G)$. Motivated by the Graph…

A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and…

Let $G$ be a connected simple graph on $n$ vertices and $m$ edges. Denote $N_{i}^{(j)}(G)$ the number of spanning subgraphs of $G$ having precisely $i$ edges and not more than $j$ connected components. The graph $G$ is \emph{strong} if…

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the…

Let $\mathcal{H}$ be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let $G$ be an arbitrary finite $\mathcal{H}$-free graph with minimum degree at least $k$. For $p \in [0,1]$, we form a $p$-random…

This paper studies critical ideals of graphs with twin vertices, which are vertices with the same neighbors. A pair of such vertices are called replicated if they are adjacent, and duplicated, otherwise. Critical ideals of graphs having…