相关论文: Cactus Trees, and Estimations of the Spectral Radi…

A cactus is a connected graph in which any two cycles have at most one common vertex. We determine the unique graph that maximizes the distance spectral radius over all cacti with fixed numbers of vertices and cycles, and thus prove a…

The subpath number of a graph G is defined as the total number of subpaths in G, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper [5], where we…

We study the Randic index for cactus graphs. It is conjectured to be bounded below by radius (for other than an even path), and it is known to obey several bounds based on diameter. We study radius and diameter for cacti then verify the…

Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral…

We prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases it can be improved.…

The square of a connected graph $G$ is obtained from $G$ by adding an edge between every pair of vertices at distance $2$. In this paper we give some upper or lower bounds for the spectral radius of the square of connected graphs, trees and…

We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain…

It is well known that the spectral radius of a tree whose maximum degree is $D$ cannot exceed $2\sqrt{D-1}$. In this paper we derive similar bounds for arbitrary planar graphs and for graphs of bounded genus. It is proved that a the…

Let $A(G)$ be the adjacent matrix and $D(G)$ the diagonal matrix of the degrees of a graph $G$, respectively. For $0 \leq \alpha \leq 1$, the $A_{\alpha}$ matrix $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ is given by Nikiforov. Clearly,…

A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…

We give upper and lower bounds on the spectral radius of a graph in terms of the number of walks. We generalize a number of known results.

In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric…

We construct an exact expression for the site percolation threshold p_c on a quasi-regular tree, and a related exact lower bound for a quasi-regular graph. Both are given by the inverse spectral radius of the appropriate Hashimoto matrix…

The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$ and $Sz_{ev}(G)=\frac{1}{2} \sum\limits_{uv \in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$…

For a graph $G = (V, E)$, a subset $F\subset V(G)$ is called an $R_k$-vertex-cut of $G$ if $G -F$ is disconnected and each vertex $u \in V(G)- F$ has at least $k$ neighbors in $G -F$. The $R_k$-vertex-connectivity of $G$, denoted by…

In this article, we obtain an upper bound for the regularity of the binomial edge ideal of a graph whose every block is either a cycle or a clique. As a consequence, we obtain an upper bound for the regularity of binomial edge ideal of a…

Let $G$ be a graph with maximum degree $\Delta$, and let $G^{\sigma}$ be an oriented graph of $G$ with skew adjacency matrix $S(G^{\sigma})$. The skew spectral radius $\rho_s(G^{\sigma})$ of $G^\sigma$ is defined as the spectral radius of…

This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $\lambda\left( G\right) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $k\geq1,$…

In this paper, we present a sharp upper bound for the spectral radius of an $n$-vertex graph without $F$-minor for sufficient large $n$, where $F$ is obtained from the complete graph $K_r$ by deleting disjointed paths. Furthermore, the…

A cactus graph is a connected graph in which every block is either an edge or a cycle. In this paper, we will examine cactus graphs where all the blocks are $3$-cycles, i.e., triangular cactus graphs, of diameter $4$. Our main focus is to…