Lexicographic polynomials of graphs and their spectra

Abstract

For a (simple) graph H and non-negative integers c0, c1,..., cd (cd ? 0), p(H) = ?dk=0 ck?Hk is the lexicographic polynomial in H of degree d, where the sum of two graphs is their join and ck?Hk is the join of ck copies of Hk. The graph Hk is the kth power of H with respect to the lexicographic product (H0 = K1). The spectrum (if H is connected and regular) and the Laplacian spectrum (in general case) of p(H) are determined in terms of the spectrum of H and ck's. Constructions of infinite families of cospectral or integral graphs are announced. [Project of the Serbian Ministry of Education, Science and Technological Development, Grant no. 174012 Grant no. 174033]

The H-join of arbitrary families of graphs

“…As an example, in [1], the spectrum of the 100-th lexicographic power of the Petersen graph, which has a gogool number (that is, 10 100 ) of vertices, was determined. With these powers, H k , in [3] the lexicographic polynomials were introduced and their spectra determined, for connected regular graphs H, in terms of the spectrum of H and the coefficients of the polynomial.…”