Lexicographic polynomials of graphs and their spectra

• Main Author: Cardoso, Domingos M.
• Publication Date: 2017
• Other Authors: Carvalho, Paula, Rama, Paula, Simic, Slobodan K., Stanic, Zoran
• Format: Article
• Language: eng
• Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
• Download full: http://hdl.handle.net/10773/18640
• topic: Spectral graph theory; Lexicographic product; Adjacency and Laplacian matrices; Cospectral graphs; Integral graphs

For a (simple) graph $H$ and non-negative integers $c_0,c_1,\ldots,c_d$ ($c_d \neq 0$), $p(H)=\sum_{k=0}^d{c_k \cdot H^k}$ is the lexicographic polynomial in $H$ of degree $d$, where the sum of two graphs is their join and $c_k \cdot H^k$ is the join of $c_k$ copies of $H^k$. The graph $H^k$ is the $k$th power of $H$ with respect to the lexicographic product ($H^0 = K_1$). The spectrum (if $H$ is regular) and the Laplacian spectrum (in general case) of $p(H)$ are determined in terms of the spectrum of $H$ and~$c_k$'s. Constructions of infinite families of cospectral or integral graphs are also announced.