A unified criterion for distinguishing graphs by their spectral radius.

Complementarity spectrum of a connected graph G, denoted by $ \Pi (G) $ Π (G) , is the set of spectral radii of all connected induced subgraphs of G. Further, G is said to be spectrally non-redundant if $ c(G) $ c (G) , the cardinality of $ \Pi (G) $ Π (G) , is equal to $ b(G) $ b (G) , 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. [ABSTRACT FROM AUTHOR]