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Register a new userSpectral characterizations of two families of nearly complete bipartite graphs
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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.
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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 $mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of $C_4$, $K_{2,t}$, $K_{3,3}$ or $K_{s,t}$ when $t>s!$.
Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined by its spectrum. In Wang~[J. Combin. Theory, Ser. B, 122 (2017):438-451], the author gave a simple arithmetic condition for a family of graphs being determined by their generalized spectra. However, the method applies only to a family of the so called emph{controllable graphs}; it fails when the graphs are non-controllable. In this paper, we introduce a class of non-controllable graphs, called emph{almost controllable graphs}, and prove that, for any pair of almost controllable graphs $G$ and $H$ that are generalized cospectral, there exist exactly two rational orthogonal matrices $Q$ with constant row sums such that $Q^{rm T}A(G)Q=A(H)$, where $A(G)$ and $A(H)$ are the adjacency matrices of $G$ and $H$, respectively. The main ingredient of the proof is a use of the Binet-Cauchy formula. As an application, we obtain a simple criterion for an almost controllable graph $G$ to be determined by its generalized spectrum, which in some sense extends the corresponding result for controllable graphs.
In this paper we give two characterizations of the $p times q$-grid graphs as co-edge-regular graphs with four distinct eigenvalues.
Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.
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