The Maximum Spectral Radius of Non-Bipartite Graphs Forbidding Short Odd Cycles

Abstract

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.

Spectral extremal graphs without intersecting triangles as a minor

“…More precisely, what is the maximum spectral radius among all n-vertex H-free graphs? In the past few decades, the problem has been investigated by many researchers for various graphs H, such as, the complete graphs [23,34,20], the complete bipartite graphs [2,24], the books and theta graphs [38], the friendship graphs [5,37,40], the intersecting odd cycles [18,11], the intersecting cliques [12], the paths and linear forests [25,10], the odd wheels [6], the quadrilateral [23,35], the hexagon [36], the short odd cycles [15,17,19], the square of a path [41], the fan graph [32]. We refer the reader to [27] for a comprehensive survey.…”

Spectral Radius of Graphs with Given Size and Odd Girth

“…In 2022, Wang [22,Theorem 5] improved slightly Theorem 4 by determining the m-edge graphs G for every m, if G is a triangle-free and non-bipartite graph with ρ(G) ⩾ √ m − 2. Very recently, by applying Cauchy's interlacing theorem of all eigenvalues, Li and Peng [10] found some forbidden induced subgraphs and presented an alternative proof of Theorem 4. Note that the unique extremal graph in Theorem 4 contains many copies of C 5 .…”

“…Note that the unique extremal graph in Theorem 4 contains many copies of C 5 . Moreover, Li and Peng [10] considered the further stability result on Theorem 4 by forbidding both C 3 and C 5 as below. Let γ(m) denote the largest root of…”

“…In this paper, we aim to generalize all the above results and confirm Conjecture 6. Unlike the techniques in [10] where the authors developed the key ideas from [11], in present paper, we will expand the ideas mainly from [25]. The odd girth of a graph is defined as the length of a shortest odd cycle.…”

Spectral extremal problem on $t$ copies of $\ell$-cycle

“…Thus we always have ex(n, F ) ≤ n 2 spex(n, F ). In recent years, the investigation on spex(n, F ) has become very popular (see [6,7,8,15,16,17,18,26,29,30,32,34] ). In this paper, we are interested in studying spex(n,tF ) for some given F. Up to now, spex(n,tF ) and its corresponding extremal graphs were studied for some special cases (see spex(n,tK 2 ) [12], spex(n,tP ℓ ) [2], spex(n,tS ℓ ) [3], spex(n,tK ℓ ) [20]).…”