Vol. 38, No. 2, 1971

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The polynomial of a non-regular digraph

William Garfield Bridges

Vol. 38 (1971), No. 2, 325–341

This paper studies the matrix equation f(Z) = D + λJ where f is a polynomial, Z a square (0,1)-matrix, D is diagonal, λ0 and J is the matrix of ones. If Z is thought of as the incidence matrix of a digraph G, the equation implies various path length properties for G. It is shown that such a graph is an amalgamation of regular subgraphs with similar path length properties. Necessary and sufficient parameter conditions on the matrix Z are given in order that it satisfy such an equation for a fixed polynomial f and all non-regular digraphs corresponding to quadratic polynomials f are found.

Mathematical Subject Classification 2000
Primary: 05C20
Received: 6 October 1970
Revised: 16 March 1971
Published: 1 August 1971
William Garfield Bridges