# Incidence Matrix

• Ravindra B. Bapat
Chapter
Part of the Universitext book series (UTX)

## Abstract

This chapter is devoted to the study of the adjacency matrix of a graph. The eigenvalues of a graph are defined to be the eigenvalues of its adjacency matrix. We first obtain the eigenvalues of the cycle and the path explicitly. A combinatorial description of the determinant of the adjacency matrix is provided. Some basic bounds involving the extreme eigenvalues of the adjacency matrix are provided, with detailed proofs. The energy of a graph, which finds applications in mathematical chemistry, is introduced. It is shown that the energy of a graph cannot be an odd integer. An elegant result of Stanley on counting directed paths is proved. In the final section we discuss trees which have a nonsingular adjacency matrix, and identify the cases when the inverse of the adjacency matrix corresponds to a graph.

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