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The characteristic polynomial of a generalized join graph

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  • Chen, Yu
  • Chen, Haiyan

Abstract

For a graph G with adjacency matrix A(G) and degree-diagonal matrix D(G), Cvetković et al introduced a bivariate polynomial ϕG(x,t)=det(xI−(A(G)−tD(G))), where I is the identity matrix. The polynomial ϕG(x, t) not only generalizes the characteristic polynomials of some well-known matrices related to G, such as the adjacency, the Laplacian matrices, but also has an elegant combinatorial interpretation as being equivalent to the Bartholdi zeta function. Let G=H[G1,G2,…,Gk] be the generalized join graph of G1,G2,…,Gk determined by graph H. In this paper, we first give a decomposition formula for ϕG(x, t). The decomposition formula provides us a new method to construct infinitely many pairs of non-regular ϕ-cospectral graphs. Then, as applications, explicit expressions for ϕG(x, t) of some special kinds of graphs are given.

Suggested Citation

  • Chen, Yu & Chen, Haiyan, 2019. "The characteristic polynomial of a generalized join graph," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 456-464.
    • Handle: RePEc:eee:apmaco:v:348:y:2019:i:c:p:456-464
    DOI: 10.1016/j.amc.2018.12.013
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