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We consider a construction using a pair of commuting regular graphs that generalizes the constructions of the middle, total, and quasitotal graphs. We derive formulae for the characteristic polynomials of the adjacency and Laplacian matrices and for the Ihara zeta function of the resulting graph. Using these formulae, we express the number of spanning trees and the Kirchhoff index of the resulting...
In this paper we consider the characteristic polynomials of not necessarily ranked posets. We do so by allowing the rank to be an arbitrary function from the poset to the nonnegative integers. We will prove two results showing that the characteristic polynomial of a poset has nonnegative integral roots. Our factorization theorems will then be used to show that any interval of a crosscut-simplicial...
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