05C50 Online Seminar: Sasmita Barik

Let G be a simple connected graph and A(G) be its adjacency matrix. A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. Furthermore, if each eigenvalue of G and its reciprocal have the same multiplicity, then G is said to have the strong reciprocal eigenvalue property. Such graphs exist and have been studied in the past. A general question remained open. Can there be a graph that has at least one zero eigenvalue and whose nonzero eigenvalues satisfy the reciprocal eigenvalue property? We first prove that there is no such nontrivial tree. Suppose that G is singular and the characteristic polynomial of G is x^{n-k} ( x^k + a_1x^{k-1} + ... + a_k ). Assume that A(G) has rank k, so that a_k\neq 0. Can we ever have |a_k|=1? The answer turns out to be negative. As an application, we prove that there is no nontrivial singular graph whose nonzero eigenvalues satisfy the reciprocal eigenvalue property. This talk is based on joint work with Debabrota Mondal, Sukanta Pati, and Kuldeep Sarma.

The slides and a recording of the talk will be shared on the main website.