A Lexicographic product for Signed Graphs

A signed graph is a pair = (G; ), where G = (V (G);E(G)) is a graph and E(G) {+1;−1} is the sign function on the edges of G. The notion of composition (also known as lexicographic product) of two signed graphs and = (H; ) already exists in literature, yet it fails to map balanced graphs onto balanced graphs. We improve the existing denition showing that our `new' signature on the lexicographic product of G and H behaves well with respect to switching equivalence. Signed regularities and some spectral properties are also discussed

ON THE LAPLACIAN SPECTRA OF PRODUCT GRAPHS

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs