Abstract
We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to -2, or 0, and determine which of these graphs are determined by their adjacency spectrum.
Highlights
In an earlier paper [3] we determined the class G of graphs with at most two adjacency eigenvalues different from ±1
The classification was motivated by the question whether the friendship graph is determined by its spectrum
We deal with the class H of graphs for which the adjacency matrix A has all but at most two eigenvalues equal to −2 or 0
Summary
In an earlier paper [3] we determined the class G of graphs with at most two adjacency eigenvalues (multiplicities included) different from ±1. We deal with the class H of graphs for which the adjacency matrix A has all but at most two eigenvalues equal to −2 or 0. The major part of our proof deals with graphs in H with two positive eigenvalues. Vm. Consider a symmetric matrix A of order n, with rows and columns partitioned according to P. (ii) The eigenvectors orthogonal to the columns of χP ; the corresponding eigenvalues of A remain unchanged if some scalar multiple of the all-one block J is added to block Ai, j for each i, j ∈ {1, . We denote the m × n all-ones matrix by Jm,n (or just J ), the all-zeros matrix by O, and the identity matrix of order n by In , or I
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