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Register a new userThe non-bipartite integral graphs with spectral radius three
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In this paper, we classify the connected non-bipartite integral graphs with spectral radius three.
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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.
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.
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.
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.
A graph H is k-common if the number of monochromatic copies of H in a k-edge-coloring of K_n is asymptotically minimized by a random coloring. For every k, we construct a connected non-bipartite k-common graph. This resolves a problem raised by Jagger, Stovicek and Thomason [Combinatorica 16 (1996), 123-141]. We also show that a graph H is k-common for every k if and only if H is Sidorenko and that H is locally k-common for every k if and only if H is locally Sidorenko.
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