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Abstract:

Given a graph G with its adjacency matrix A, consider the matrix A(x, y) in which the 1s are replaced by the indeterminate x and 0s (other than the diagonals) are replaced by y. The L-polynomial of G is defined as: LG(x, y, λ): = det(A(x, y) − λI). This polynomial is a natural generalization of the standard characteristic polynomial of a graph. In this note we characterize graphs which have the same L-polynomial. The answer is rather simple: Two graphs G and H have the same L-polynomial if and only if- G and H are co-spectral and Gc and Hc are co-spectral. (Here Gc (resp. Hc) is the complement of G (resp. H).) 1

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