Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2023; 60(5): 959-986

Online first article August 18, 2023      Printed September 1, 2023

Copyright © The Korean Mathematical Society.

Forbidden theta graph, bounded spectral radius and size of non-bipartite graphs

Shuchao Li, Wanting Sun, Wei Wei

Central China Normal University; Shandong University; Shanghai University of Engineering Science


Zhai and Lin recently proved that if $G$ is an $n$-vertex connected $\theta(1, 2, r+1)$-free graph, then for odd $r$ and $n \geqslant 10r$, or for even $r$ and $n \geqslant 7r$, one has $\rho(G) \le \sqrt{\lfloor\frac{n^2}{4}\rfloor}$, and equality holds if and only if $G$ is $K_{\lceil\frac{n}{2}\rceil, \lfloor\frac{n}{2}\rfloor}$. In this paper, for large enough $n$, we prove a sharp upper bound for the spectral radius in an $n$-vertex $H$-free non-bipartite graph, where $H$ is $\theta(1, 2, 3)$ or $\theta(1, 2, 4)$, and we characterize all the extremal graphs. Furthermore, for $n \geqslant 137$, we determine the maximum number of edges in an $n$-vertex $\theta(1, 2, 4)$-free non-bipartite graph and characterize the unique extremal graph.

Keywords: Spectral radius, size, (spectral) Tur\'{a}n type problem, theta-free graph

MSC numbers: 05C50, 05C35

Supported by: Financially supported by the National Natural Science Foundation of China (Grant Nos. 12171190, 11671164.

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