Brualdi-Hoffman-Tur\'{a}n problem of the gem

TL;DR

This paper fully solves the spectral radius maximization problem for gem-free graphs of any size, extending previous partial results for odd sizes and providing a complete characterization.

Contribution

It provides a complete solution to the Brualdi-Hoffman-Turán problem for the gem, including cases not previously addressed.

Findings

Determined the maximum spectral radius for gem-free graphs of all sizes.

Extended previous results from odd sizes to all sizes.

Established the extremal graphs achieving the maximum spectral radius.

Abstract

A graph is said to be $F$-free if it does not contain $F$ as a subgraph. Brualdi-Hoffman-Tur\'{a}n problem seeks to determine the maximum spectral radius of an $F$-free graph with given size. The gem consists of a path on $4$ vertices, along with an additional vertex that is adjacent to every vertex of the path. Concerning Brualdi-Hoffman-Tur\'{a}n problem of the gem, when the size is odd, Zhang and Wang [Discrete Math. 347 (2024) 114171] and Yu, Li and Peng [arXiv:2404. 03423] solved it. In this paper, we completely solve the Brualdi-Hoffman-Tur\'{a}n problem type problem of the gem.