Spectra and elementary cycles of the digraphs with unique paths of fixed length

Abstract

A digraph G, whose adjacency matrix A satisfies A k=J n−I n , where J n is the n×n matrix of all ones, is called a digraph with unique paths of fixed length k, or simply a UPFL- k digraph. We prove that all the UPFL- k digraphs of the same order are co-spectral and have the same number of elementary cycles of length l for each l⩽k. We also provide some techniques helpful for computing the spectrum and the numbers of short elementary cycles of a UPFL digraph, including the determination of the numbers of reentrant paths of every fixed length in a UPFL digraph. At the end of the paper we point out an interesting relation between the number of elementary cycles of the UPFL digraphs and the number of circular sequences with equal length and period. Our theorems generalize corresponding results of Lam and Van Lint.