Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we show that, there is a number $\eta(m,k)>\sqrt{m-k+3}$ such that every non bipartite graph $G$ with size $m$ and spectral radius $\rho\ge \eta(m,k)$ must contain an odd cycle of length less than $k$ unless $m$ is odd and $G\cong SK_{k,m}$, which is the graph obtained by subdividing an edge $k-2$ times of the complete bipartite graph $K_{2,\frac{m-k+2}{2}}$. This result implies the main results of Zhai and Shu [Discrete Math. 345 (2022)] and settles a conjecture of Li and Peng [The Electronic J. Combin. 29 (4) (2022)] as well.
@article{10_37236_11720,
author = {Zhenzhen Lou and Lu Lu and Xueyi Huang},
title = {Spectral radius of graphs with given size and odd girth},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {1},
doi = {10.37236/11720},
zbl = {1535.05178},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11720/}
}
TY - JOUR
AU - Zhenzhen Lou
AU - Lu Lu
AU - Xueyi Huang
TI - Spectral radius of graphs with given size and odd girth
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11720/
DO - 10.37236/11720
ID - 10_37236_11720
ER -
%0 Journal Article
%A Zhenzhen Lou
%A Lu Lu
%A Xueyi Huang
%T Spectral radius of graphs with given size and odd girth
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11720/
%R 10.37236/11720
%F 10_37236_11720
Zhenzhen Lou; Lu Lu; Xueyi Huang. Spectral radius of graphs with given size and odd girth. The electronic journal of combinatorics, Tome 31 (2024) no. 1. doi: 10.37236/11720
