相关论文: Bounds on graph eigenvalues II

The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest…

The $p$-spectral radius of a graph $G\ $of order $n$ is defined for any real number $p\geq1$ as \[ \lambda^{\left( p\right) }\left( G\right) =\max\left\{ 2\sum_{\{i,j\}\in E\left( G\right) \ }x_{i}x_{j}:x_{1},\ldots,x_{n}\in\mathbb{R}\text{…

Given a graph $F$, the expansion $F^{(r)}$ of $F$ is defined as the $r$-uniform hypergraph obtained from $F$ by adding a set of $(r-2)$ distinct new vertices to each edge of $F$. In this paper, we investigate spectral stability results for…

Let G be a graph of given order and mu(G) be the largest eigenvalue of its adjacency matrix. We give conditions on mu(G) that imply Hamiltonicity of G and of its complement.

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality…

Let $H$ be a graph. We show that if $r$ is large enough as a function of $H$, then the $r$-partite Tur\'an graph maximizes the number of copies of $H$ among all $K_{r+1}$-free graphs on a given number of vertices. This confirms a conjecture…

The spectrum of a graph is closely related to many graph parameters. In particular, the spectral gap of a regular graph which is the difference between its valency and second eigenvalue, is widely seen an algebraic measure of connectivity…

The Tur\'an problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $\Omega(n^{2-1/s})$ edges…

The classical spectral Tur\'{a}n problem is to determine the maximum spectral radius of an $\mathcal{F}$-free graph of order $n$. Zhai and Wang [Linear Algebra Appl, 437 (2012) 1641-1647] determined the maximum spectral radius of…

In this paper, the extremal graph with the maximum RRD-value among all the graphs of order n with given cut vertices or edges is characterized. In addition, an upper bounds on the reciprocal degree distance in terms of the number of cut…

Characterized are all simple undirected graphs $G$ such that any real symmetric matrix that has graph $G$ has no eigenvalues of multiplicity more than 2. All such graphs are partial 2-trees (and this follows from a result for rather general…

The Turan number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. We determine the Turan number and find the unique extremal graph for forests consisting of paths when n is…

The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum…

Let $G$ be a graph with $n$ vertices and $\lambda_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal…

Let $G$ be a connected uniform hypergraphs with maximum degree $\Delta$, spectral radius $\lambda$ and minimum H-eigenvalue $\mu$. In this paper, we give some lower bounds for $\Delta-\lambda$, which extend the result of [S.M. Cioab\u{a},…

The edge-spectral-Tur\'an type problem is also called the Brualdi-Hoffman-Tur\'an type problem, which is a central topic in spectral graph theory, seeking to determine the maximum spectral radius $\lambda(G)$ of an $F$-free graph $G$ with…

A signed graph $(G,\sigma)$ is a graph $G$ together with an assignment $\sigma$ of either a positive sign or a negative sign to each edge. A signed graph is unbalanced if it contains a cycle with odd number of negative edges. The spectral…

A signed graph is a graph in which every edge carries a $+$ or a $-$ sign. In this paper, we determine the signed graphs with maximum spectral radius among all unbalanced signed graphs with fixed order that contain neither negative…

Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove…

A graph on $2k+1$ vertices consisting of $k$ triangles which intersect in exactly one common vertex is called a $k$-fan and denoted by $F_k$. This paper aims to determine the graphs of order $n$ that have the maximum (adjacency) spectral…