Related papers: New results on eigenvalues and degree deviation

Collatz and Sinogowitz had proposed to measure the departure of a graph $G$ from regularity by the difference of the (adjacency) spectral radius and the average degree: $\epsilon(G)=\rho(G)-\frac{2m}{n}$. We give here new lower bounds on…

Let $G$ be a graph with $n$ vertices and $m$ edges. The spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of the adjacency matrix of $G$. As is well known, $\rho(G)\geq\frac{2m}{n}$ with equality if and only if $G$ is regular. To…

For a finite, simple, and undirected graph $G$ with $n$ vertices and average degree $d$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-d\right|$. Provided that $G$ has largest eigenvalue $\lambda$,…

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a…

Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give…

We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix.…

Let $G$ be a graph attaining the maximum spectral radius among all connected nonregular graphs of order $n$ with maximum degree $\Delta$. Let $\lambda_1(G)$ be the spectral radius of $G$. A nice conjecture due to Liu, Shen and Wang [On the…

Given a graph G of order n and size m, let s(G)= sum|d(u)-2m/n|, where the sum is taken over all vertices u of G. We investigate upper and lower bounds on eigenvalues of G in terms of s(G).

In this study we are interested mainly in investigating the relations between two graph irregularity measures which are widely used for structural irregularity characterization of connected graphs. Our study is focused on the comparison and…

The spectral radius {\rho}(G) of a digraph G is the maximum modulus of the eigenvalues of its adjacency matrix. We present bounds on {\rho}(G) that are often tighter and are applicable to a larger class of digraphs than previously reported…

Brualdi and Hoffman proposed a well-known problem of determining the graph with maximum adjacency spectral radius among all graphs with given size $m$. Early work by Friedland and Stanley addressed some specific cases. This problem was…

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of…

Let $G$ be a graph with minimum degree $\delta$. The spectral radius of $G$, denoted by $\rho(G)$, is the largest eigenvalue of the adjacency matrix of $G$. In this note we mainly prove the following two results. (1) Let $G$ be a graph on…

Let G be an n-vertex graph with m edges. The degree deviation measure of G is defined as s(G)=sum v in V(G)|degG(v)-(2m/n)|, where n and m are the number of vertices and edges of G, respectively. The aim of this paper is to prove the…

Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of…

Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $\rho$ of $G$ is the maximum average distance from a vertex to all others and $\partial_1\geq\cdots\geq \partial_n$ are the distance eigenvalues of $G$. In \cite{AH},…

Let $G$ be a random graph on the vertex set $\{1,2,..., n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probability $p_{ij}$ for $\{i,j\}$ being an edge in $G$ is not assumed to be equal.…

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov [Appl. Anal. Discrete Math., 11 (2017) 81--107] defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real…

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…