Perturbation of the largest matching root of hypergraphs

TL;DR

This paper studies how the largest matching root of hypergraphs changes under perturbations, identifying extremal hypergraphs and extending classical graph transformation results.

Contribution

It characterizes hypergraphs with maximum largest matching root among certain classes and proves that the shifting operation does not decrease this root.

Findings

Largest matching root is maximized by specific hypergraph structures.

Shifting operation does not decrease the largest matching root.

Extension of classical graph results to 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 Erd\H{o}s, Ko and Rado, does not decrease the largest matching root. This result extends a classical result by Csikv\'ari (Electron. J. Combin. {\bf 18} (2011) $\#$P182) stating that the Kelmans transformation does not decrease the largest matching root of graphs.