SPECTRA OF GRAPH OPERATIONS BASED ON SPLITTING GRAPH

Abstract

The splitting graph $ SP(G) $ of a graph $ G $ is the graph obtained from $ G $ by taking a new vertex $ u' $ for each $ u \in V(G) $ and joining $ u' $ to all vertices of $ G $ adjacent to $ u $. For a connected regular graph $ G_1 $ and an arbitrary regular graph $ G_2 $, we determine the adjacency (respectively, Laplacian and signless Laplacian) spectra of two types of graph operations on $ G_1 $ and $ G_2 $ involving the $ SP $-graph of $ G_1 $. Moreover, applying these results we construct some non-regular simultaneous cospectral graphs for the adjacency, Laplacian and signless Laplacian matrices, and compute the Kirchhoff index and the number of spanning trees of the newly constructed graphs.