On the eigenvalues of closed shadow graphs

Abstract

Abstract The energy and d-energy of a graph is the sum of the absolute values of the graphs’ adjacency and distance matrix respectively. In 2017, S. Vaidya and K. Popat determined the energy of a shadow graph of a connected graph. In this study we determine the eigenvalues and d-eigenvalues of a closed shadow graph. Thus, determining also the energy and distance energy of a closed shadow graph. As a consequence, the closed shadow graph of G is integral whenever G is integral, that is the eigenvalues of the adjacency matrix of the graph are all integers.