Research On Singularly Cospectral And Equienergetic Graphs

The spectral graph theory of is a key area of research in algebraic graph theory,which applies the methods and techniques of combinatorial matrix theory to study the structure,combination,topology and chemical indicators of graph through the algebraic properties of graph matrix such as adjacency matrix and Laplacian matrix.Two graphs are called singularly cospectral if they have the same nonzero singular values with the same multiplicities.The graph energy is defined as the sum of the absolute values of the eigenvalues of adjacency matrix,which is a topical problem in chemical graph theory.This paper focuses on the construction and properties of singularly cospectral and equienergetic graphs.The main contents are as follows:Firstly,the definition of t-join corona graph is given.Based on the regular singularly cospectral graph of smaller order,we construct infinite pairs of singularly cospectral graphs,using the determinant calculation technique of block matrix and Hoffman theorem.For irregular singularly cospectral graphs,an infinite number of pairs of singularly cospectral graphs are also constructed using the t-shadow corona graph operation.The above results extend the conclusions of Indulal and Vijayakumar.Secondly,we study the spectral properties of Kronecker product and Cartesian product of singularly cospectral graphs.We prove that a pair of singularly cospectral graph and a pair of cospectral bipartite graph is cospectral after Kronecker product and singularly cospectral after Cartesian product,and a pair of singularly cospectral graph and a pair of cospectral non-bipartite graph is singularly cospectral after Kronecker product.Based on this property,it is shown that m-shadow graph and m-splitting graph of singularly cospectral graphs are still singularly cospectral.Finally,it is proved that a class of commutative singular cospectral graphs and he same bipartite graph are isomorphic after Kronecker product.Then the energy of iterated line graphs of the singularly cospectral graphs is studied by programming.Numerical experiments show that for some singularly cospectral graphs,they become equienergetic graphs after 1 to 4 line graph iterations.Finally,order 7 and 8connected singularly cospectral and their spectra are enumerated,and order 9 regular connected singular singularly cospectral is presented.