Construction of L-equienergetic graphs using some graph operations

Abstract

For a graph G with n vertices and m edges, the eigenvalues of its adjacency matrix A (G) are known as eigenvalues of G. The sum of absolute values of eigenvalues of G is called the energy of G. The Laplacian matrix of G is defined as L (G)= D (G)− A (G) where D (G) is the diagonal matrix with (i, j) th entry is the degree of vertex vi. The collection of eigenvalues of L (G) with their multiplicities is called spectra of L (G). If μ 1, μ 2,⋯, μ n are the eigenvalues of L (G) then the Laplacian energy LE (G) of G is defined as LE (G)=∑ i= 1 n| μ i− 2 mn|. It is always interesting and challenging as well to investigate the graphs which are L-equienergetic but L-noncopectral as L-cospectral graphs are obviously L-equienergetic. We have devised a method to construct L-equienergetic graphs which are L-noncospectral.