Related papers: Enumeration of copermanental graphs

We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of…

In this paper, we focus on the study of immanantal polynomials for linear combination matrices composed of the degree matrix and adjacency matrix of a graph. First, applying the concept of vertex orientation for general graphs, we provide a…

Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite…

A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$…

In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space - in particular, to every finite simple connected graph $G$ - and showed it has several attractive properties. First, it is…

We introduce the strong alliance polynomial of a graph. The strong alliance polynomial of a graph $G$ with order n and strong defensive alliance number $a(G)$ is the polynomial $a(G;x):=\sum_{i=a(G)}^{n}\, a_i(G)\ x^i$, where $a_{k}(G)$ is…

The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…

A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = \alpha A + \beta I + \gamma J + \delta D$ with $\alpha \neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the…

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,…

We show that the number of $k$-matching in a given undirected graph $G$ is equal to the number of perfect matching of the corresponding graph $G_k$ on an even number of vertices divided by a suitable factor. If $G$ is bipartite then one can…

Let $G$ be a connected graph on $n$ vertices with adjacency matrix $A_G$. Associated to $G$ is a polynomial $d_G(x_1,\dots, x_n)$ of degree $n$ in $n$ variables, obtained as the determinant of the matrix $M_G(x_1,\dots,x_n)$, where…

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…

As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex $u$ induces the same regular partition around $u$, where all vertices of each cell…

A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for…

Let $\Phi(x,y)\in\mathbb{C}[x,y]$ be a symmetric polynomial of partial degree $d$. The graph $G(\Phi)$ is defined by taking $\mathbb{C}$ as set of vertices and the points of $\mathbb{V}(\Phi(x,y))$ as edges. We study the following problem:…

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\mu$, we define the polynomial $$P_\mu(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}\,\mu^{\ell(\sigma)}\; ,$$ as the $\mu$-permanent of $A$, where $\ell(\sigma)$…

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 domination sets of each cardinality in $G$, and its…

We give a method of generating strongly polynomial sequences of graphs, i.e., sequences $(H_{\mathbf{k}})$ indexed by a multivariate parameter $\mathbf{k}=(k_1,\ldots, k_h)$ such that, for each fixed graph $G$, there is a multivariate…

Assumed to be undirected, simple, and connected are all of the graphs in this study, and adjacency matrix $A$ serves as the associated matrix. In this paper we show that it is possible to relate a creation sequence for a type of cographs…

In this paper, we study the independence polynomial $P_G(x)$ of a finite simple graph $G$, with emphasis on the evaluation at $x=-1$, symmetry, and its connection with the $h$-polynomial of the edge ideal of $G$. For big star graphs, we…