相关论文: On the largest real root of independence polynomia…

Given $n \geq 1$, we study the existence of a tree on $n$ vertices whose independence polynomial is symmetric and unimodal as well as the existence of a symmetric and unimodal independence polynomial of degree $n$ of a tree.

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…

Let $G$ be a graph of order $n$ and let $k\in\{1,\ldots,n-1\}$. The $k$-token graph $F_k(G)$ of $G$, is the graph whose vertices are the $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference…

An independent set in a simple graph $G$ is a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$, denoted by $I_G$ is defined as $1 + a_1 z + a_2 z^2+\cdots+a_d z^{d}$, where $a_i$ denotes the number of…

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 independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the…

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). Alavi, Malde, Schwenk…

In [1] Peters and Regts confirmed a conjecture by Sokal by showing that for every $\Delta \in \mathbb{Z}_{\geq 3}$ there exists a complex neighborhood of the interval $\left[0, \frac{\left(\Delta - 1\right)^{\Delta -…

The adjoint polynomial of $G$ is \[h(G,x)=\sum_{k=1}^n(-1)^{n-k}a_k(G)x^k,\] where $a_k(G)$ denotes the number of ways one can cover all vertices of the graph $G$ by exactly $k$ disjoint cliques of $G$. In this paper we show the the adjoint…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A subset $I$ of $V(G)$ is an independent vertex subset if no two vertices in $I$ are adjacent in $G$. We study the number, $\sigma_1(G)$, of all subsets of $v(G)$ that contain…

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with…

The independence equivalence class of a graph $G$ is the set of graphs that have the same independence polynomial as $G$. A graph whose independence equivalence class contains only itself, up to isomorphism, is independence unique. Beaton,…

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph…

The Wiener polynomial of a connected graph $G$ is defined as $W(G;x)=\sum x^{d(u,v)}$, where $d(u,v)$ denotes the distance between $u$ and $v$, and the sum is taken over all unordered pairs of distinct vertices of $G$. We examine the nature…

For each uniformity $k \geq 3$, we construct $k$-uniform linear hypergraphs $G$ with arbitrarily large maximum degree $\Delta$ whose independence polynomial $Z_G$ has a root $\lambda$ with $\lvert\lambda\rvert = O\left(\frac{\log…

For a graph $G$ with $n$ vertices and a positive integer $k \leq n$, let $s_k(G)$ be the number of subtrees (subgraphs that are trees, not necessarily induced) of $G$ with $k$ vertices. The subtree polynomial of $G$ is $S(G;x) =…

The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathscr{I}(G)$, is the graph whose vertices correspond to the…

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if they have equivalent independence polynomials. We extend…

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…

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…