The spectral radius of graphs with no intersecting odd cycles

Abstract

Let $H_{s,t_1,\dots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\dots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph Theory 89 (1) (2018) 26--39] determined independently the maximum number of edges in an $n$-vertex graph that does not contain $H_{s,t_1,\dots ,t_k}$ as a subgraph. In this paper, we determine the graphs of order $n$ that attain the maximum spectral radius among all graphs containing no $H_{s,t_1,\dots ,t_k}$ for $n$ large enough.

25 pages. This is the Journal Version. The problem raised at the end of our paper was recently solved by Chen, Liu and Zhang; see arXiv:2108.03895. The extremal spectral problem involving the intersecting cliques was also solved in another paper; see the joint work arXiv:2108.03587v2. arXiv admin note: text overlap with arXiv:1911.13082 by other authors

References in corpus (2)

• A proof of the stability of extremal graphs, Simonovits' stability from Szemerédi's regularity
• The spectral radius of graphs with no intersecting triangles

Cited by in corpus (4)

• Spectral extremal graphs for the bowtie
• Refinement on spectral Turán's theorem
• The signless Laplacian spectral radius of graphs without intersecting odd cycles
• Spectral extrema of $K_{s,t}$-minor free graphs--On a conjecture of M. Tait