Abstract
We replace certain edges of a directed graph by chains and consider the effect on the spectrum of the graph. We show that the spectral radius decreases monotonically with the expansion and that, for a strongly connected graph that is not a single cycle, the spectral radius decreases strictly monotonically with the expansion. We also give a limiting formula for the spectral radius of the expanded graph when the lengths of some chains replacing the original edges tend to infinity. Our proofs depend on the construction of auxiliary nonnegative matrices of the same size and with the same support as the original adjacency matrix.
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