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Let k, p, q be positive integers with k < p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph Kp,q of bipartition orders p and q by deleting k edges is attained when the deleting edges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences.
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Let $G$ denote a bipartite graph with $e$ edges without isolated vertices. It was known that the spectral radius of $G$ is at most the square root of $e$, and the upper bound is attained if and only if $G$ is a complete bipartite graph. Suppose that $G$ is not a complete bipartite graph, and $e-1$ and $e+1$ are not twin primes. We determine the maximal spectral radius of $G$. As a byproduct of our study, we obtain a spectral characterization of a pair $(e-1, e+1)$ of integers to be a pair of twin primes.
In this paper, we classify the connected non-bipartite integral graphs with spectral radius three.
Let $G$ be a simple graph with vertex set $V(G) = {v_1 ,v_2 ,cdots ,v_n}$. The Harary matrix $RD(G)$ of $G$, which is initially called the reciprocal distance matrix, is an $n times n$ matrix whose $(i,j)$-entry is equal to $frac{1}{d_{ij}}$ if $i ot=j$ and $0$ otherwise, where $d_{ij}$ is the distance of $v_i$ and $v_j$ in $G$. In this paper, we characterize graphs with maximum spectral radius of Harary matrix in three classes of simple connected graphs with $n$ vertices: graphs with fixed matching number, bipartite graphs with fixed matching number, and graphs with given number of cut edges, respectively.
It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.
The odd wheel $W_{2k+1}$ is the graph formed by joining a vertex to a cycle of length $2k$. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an $n$-vertex graph that does not contain $W_{2k+1}$. We determine the structure of the spectral extremal graphs for all $kgeq 2, k otin {4,5}$. When $k=2$, we show that these spectral extremal graphs are among the Tur{a}n-extremal graphs on $n$ vertices that do not contain $W_{2k+1}$ and have the maximum number of edges, but when $kgeq 9$, we show that the family of spectral extremal graphs and the family of Tur{a}n-extremal graphs are disjoint.
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