Spectra of total graphs.

The total graph T (G) of a graph G has vertex set V (T (G)) = V (G) ∪ E (G) , and two vertices in T (G) are adjacent if and only if their corresponding elements are either adjacent or incident in G. The total graph operation can be used to generate large dense graphs (networks). In this paper, some spectral properties of total graphs are studied. We give expressions for the number of eigenvalues of T (G) belong to the interval (− 2 , ∞) and (− ∞ , − 2) , and use the expressions to derive a lower bound on the clique partition number of T (G). Expressions for the multiplicity and the eigenspace of eigenvalue − 2 of T (G) are obtained. We also give a formula for the characteristic polynomial of T (G) in terms of the adjacency matrix and signless Laplacian matrix of G , and derive some properties for the Perron vector of T (G). [ABSTRACT FROM AUTHOR]