相关论文: Spectral moments of trees with given degree sequen…

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{i}$ the degree of vertex $v_{i}$. Let $f(x, y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_{f}(G)$ of a graph $G$ is a square matrix, where the…

The sigma-irregularity index $\sigma(G) = \sum_{uv \in E(G)} (d_G(u) - d_G(v))^2$ measures the total degree imbalance along the edges of a graph. We study extremal problems for $\sigma(T)$ within the class of trees of fixed order $n$ and…

Let $G_S$ be a graph with loops attached at each vertex in $S \subseteq V(G).$ In this article, we develop exact formulae for the number of closed $3$- and $4$-walks on $G_S$ in terms of vertex degrees and certain elementary subgraphs of…

A graph $G$ is a $k$-leaf power if there exists a tree $T$ whose leaf set is $V(G)$, and such that $uv \in E(G)$ if and only if the distance between $u$ and $v$ in $T$ is at most $k$. The graph classes of $k$-leaf powers have several…

An equitable tree-$k$-coloring of a graph is a vertex $k$-coloring such that each color class induces a forest and the size of any two color classes differ by at most one. In this work, we show that every interval graph $G$ has an equitable…

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $\Delta(G)$ the maximum degree of a graph $G$ and let $\rho(G)$ be the spectral radius of $G$. In this article we present a lower…

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors…

Let $k$ and $n$ be integers such that $1\leq k \leq n-1$, and let $G$ be a simple graph of order $n$. 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 $F_k(G)$…

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, an $S$-Steiner tree $T$ is a subgraph of $G$ that is a tree with $S\subseteq V(T)$. Two $S$-Steiner trees $T$ and $T'$ are internally disjoint (resp. edge-disjoint) if…

A graph is maximal $k$-degenerate if each induced subgraph has a vertex of degree at most $k$ and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal…

We study the problem of maximizing the number of full degree vertices in a spanning tree $T$ of a graph $G$; that is, the number of vertices whose degree in $T$ equals its degree in $G$. In cubic graphs, this problem is equivalent to…

Let $\mathcal G$ be a separable family of graphs. Then for all positive constants $\epsilon$ and $\Delta$ and for every sufficiently large integer $n$, every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum degree…

Let $k$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\delta$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge…

For a graph $G$, let $\tau(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $\tau(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for…

The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree…

We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local tree-like structure), exact equations are derived. These equations are generalized to the…

The isolation number of a graph $G$ (also called the vertex-edge domination number of $G$), denoted by $\iota(G)$, is the size of a smallest subset $D$ of the vertex set $V(G)$ of $G$ such that $G-N[D]$ (the graph obtained by deleting the…

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of $G$. An $n$-vertex graph is said to be hypoenergetic if $E(G)<n$ and strongly hypoenergetic if $E(G)<n-1$. In this paper, we consider…

Let $G$ be a simple graph of order $n$, let $\lambda_1(G),\lambda_2(G),...,\lambda_n(G)$ be the eigenvalues of the adjacency matrix of $G$. The Esrada index of $G$ is defined as $EE(G)=\sum_{i=1}^{n}e^{\lambda_i(G)}$. In this paper we…

A tree $T$ in an edge-colored graph is a \emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2\leq k\leq n$. For a vertex set $S\subseteq…