相关论文: Very well-covered graphs with log-concave independ…

If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and…

If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Let $a$ be the size of a…

A graph $G$ is well-covered if all its maximal stable sets have the same size, denoted by alpha(G) (M. D. Plummer, 1970). If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$…

If for any k the k-th coefficient of a polynomial I(G;x)is equal to the number of stable sets of cardinality k in graph G, then it is called the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered…

A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer, 1970). If G is a well-covered graph, has at least two vertices, and G-v is well-covered for every vertex v, then G is a 1-well-covered graph…

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…

The number of stable sets of cardinality $k$ in graph $G$ is the $k$-th coefficient of the independence polynomial of $G$ (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph, its independence…

An independent set in a graph is a collection of vertices that are not adjacent to each other. The cardinality of the largest independent set in $G$ is represented by $\alpha(G)$. The independence polynomial of a graph $G = (V, E)$ was…

Let $I(G;x)$ denote the independence polynomial of a graph $G$. In this paper we study the unimodality properties of $I(G;x)$ for some composite graphs $G$. Given two graphs $G_1$ and $G_2$, let $G_1[G_2]$ denote the lexicographic product…

An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $\alpha{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for…

The independence polynomial $I(G, x)$ of a graph $G$ is the polynomial in variable $x$ in which the coefficient $a_n$ on $x^n$ gives the number of independent subsets $S \subseteq V(G)$ of vertices of $G$ such that $|S| = n$. $I(G, x)$ is…

If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove…

A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_{k} is the number of independent sets of cardinality k in a graph G,…

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of dominating sets of each cardinality in $G$. In…

For some positive integer $m$, a real polynomial $P(x)=\sum\limits_{k=0}^ma_kx^k$ with $a_k\geqslant 0$ is called log-concave (resp. ultra log-concave) if $a_k^2\geqslant a_{k-1}a_{k+1}$ (resp. $a_k^2\geqslant…

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Alavi, Erd\H{o}s, Malde and Schwenk made the conjecture that if $G$ is a tree then the independent set sequence $\{i_t(G)\}_{t\geq 0}$ of $G$ is unimodal; Levit and…

An independent set in a graph is a set of pairwise non-adjacent vertices, and a(G) is the size of a maximum independent set in the graph G. If s_{k} is the number of independent sets of cardinality k in G, then…

We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as…

An independent set of size $k$ in a finite undirected graph $G$ is a set of $k$ vertices of the graph, no two of which are connected by an edge. Let $x_{k}(G)$ be the number of independent sets of size $k$ in the graph $G$ and let…

The independence polynomial of a graph $G$, denoted $I(G,x)$, is the generating polynomial for the number of independent sets of each size. The roots of $I(G,x)$ are called the \textit{independence roots} of $G$. It is known that for every…