Perturbation of the largest matching root of hypergraphs

Abstract

The largest matching root of a $k$-graph is the largest real root of its matching polynomial, which is equal to the maximum modulus of all the zeros of the matching polynomial. In this paper, we investigate the perturbation of the largest matching root of $k$-graphs. We determine all $k$-graphs whose largest matching root attains the maximum among all $k$-cacti and linear $k$-cacti with a given number of cycles and edges, where a $k$-cactus is a $k$-graph in which every two distinct cycles have at most one vertex in common. To achieve this, we prove that the celebrated shifting operation of $k$-graphs, introduced by Erdős, Ko and Rado, does not decrease the largest matching root. This result extends a classical result by Csikvári (Electron. J. Combin. {\bf 18} (2011) $\#$P182) stating that the Kelmans transformation does not decrease the largest matching root of graphs.