A unified criterion for distinguishing graphs by their spectral radius

Abstract

Complementarity spectrum of a connected graph G, denoted by (Formula presented.), is the set of spectral radii of all connected induced subgraphs of G. Further, G is said to be spectrally non-redundant if (Formula presented.), the cardinality of (Formula presented.), is equal to (Formula presented.), the number of all non-isomorphic induced subgraphs of G. In this paper, we give a sufficient condition for a family of graphs to be spectrally non-redundant. Using this criterion, we show that several infinite families of graphs are spectrally non-redundant. Moreover, we apply the same condition to distinguish graphs by their spectral radius, which illustrates the main reason for associating a graph with its complementarity spectrum.