Related papers: Clique Polynomials and Chordal Graphs

For a tree $T$, the subtree polynomial of $T$ is the generating polynomial for the number of subtrees of $T$. We show that the complex roots of the subtree polynomial are contained in the disk $\left\{z\in\mathbb{C}\colon\ |z|\leq…

We prove that for every tree $T$ which is not an edge, for almost every graph $G$ which does not contain $T$ as an induced subgraph, $V(G)$ has a partition into $\alpha(T)-1$ parts certifying this fact. Each part induces a graph which is…

For each nonnegative integer $i$, let $a_i$ be the number of $i$-subsets of $V(G)$ that induce an acyclic subgraph of a given graph $G$. We define $A(G,x) = \sum_{i \geq 0} a_i x^i$ (the generating function for $a_i$) to be the acyclic…

Let $G$ be a simple graph of order $n$ and size $m$. The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial…

The alliance polynomial of a graph $G$ with order $n$ and maximum degree $\Delta$ is the polynomial $A(G; x) = \sum_{k=-\Delta}^{\Delta} A_{k}(G) \, x^{n+k}$, where $A_{k}(G)$ is the number of exact defensive $k$-alliances in $G$. We obtain…

A graph $G = (V, E)$ is \emph{partitionable} if there exists a partition $\{A, B\}$ of $V$ such that $A$ induces a disjoint union of cliques and $B$ induces a triangle-free graph. In this paper we investigate the computational complexity of…

A graph is unichord free if it does not contain a cycle with exactly one chord as its subgraph. In [3], it is shown that a graph is unichord free if and only if every minimal vertex separator is a stable set. In this paper, we first show…

The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu…

We prove for every graph H there exists a>0 such that, for every graph G with at least two vertices, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least a|G| neighbours, or there are two disjoint…

Let R be an Artinian ring and G be the compressed zero-divisor graph associated to R. The question of when the clique number of compressed zero-divisor graphs is finite was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S.…

Let $G$ be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x) =\sum d(G, i)x^i, where d(G,i) is the number of dominating sets of G of size i. Every root of D(G,x) is called the domination root of G. It is…

An ordered graph $G_<$ is a graph with a total ordering $<$ on its vertex set. A monotone path of length $k$ is a sequence of vertices $v_1<v_2<\ldots<v_k$ such that $v_iv_{j}$ is an edge of $G_<$ if and only if $|j-i|=1$. A bi-clique of…

In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal…

A graph $G$ is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function $w$ is defined on its vertices. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight.…

In this paper we calculate the average number of cliques in random scale-free networks. We consider first the hidden variable ensemble and subsequently the Molloy Reed ensemble. In both cases we find that cliques, i.e. fully connected…

The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form…

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains…

Let $B$ and $R$ be two simple graphs with vertex set $V$, and let $G(B,R)$ be the simple graph with vertex set $V$, in which two vertices are adjacent if they are adjacent in at least one of $B$ and $R$. For $X \subseteq V$, we denote by…

For a finite group $G$, we define the inclusion graph of subgroups of $G$, denoted by $\mathcal I(G)$, is a graph having all the proper subgroups of $G$ as its vertices and two distinct vertices $H$ and $K$ in $\mathcal I(G)$ are adjacent…

In this paper, we study graphs whose matching polynomial have only integer zeros. A graph is matching integral if the zeros of its matching polynomial are all integers. We characterize all matching integral traceable graphs.. We show that…