A spectral extremal problem on non-bipartite triangle-free graphs

Abstract

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $λ(G)\le \sqrt{m}$, where the equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $λ_1^2(G) + λ_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $λ(G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $λ(G) \le β(m)$, where $β(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph.

Figures (7)

• Figure 1: $m$ is odd and the graph $SK_{2,\frac{m-1}{2}}$.
• Figure 2: Extremal graphs in Theorem \ref{['thm-main']}.
• Figure 3: The structure of $G$ when $|V_1|=1$ or $|V_1|=3$.
• Figure :
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Theorems & Definitions (33)

• Theorem 1.1: Nosal Nosal1970, Nikiforov Niki2002cpcNiki2009jctb
• Conjecture 1.2: Bollobás--Nikiforov, 2007
• Theorem 1.3: Lin--Ning--Wu, 2021
• Theorem 1.4: Zhai--Shu, 2022
• Definition 1.6: Spectral extremal graphs
• Theorem 1.7: Main result
• Lemma 2.1
• proof
• Lemma 2.2
• proof