Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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FORBIDDEN THETA GRAPH, BOUNDED SPECTRAL RADIUS AND SIZE OF NON-BIPARTITE GRAPHS

  • Shuchao Li (Hubei Key Laboratory of Mathematical Science and Faculty of Mathematics and Statistics Central China Normal University) ;
  • Wanting Sun (Data Science Institute Shandong University) ;
  • Wei Wei (Center of Intelligent Computing and Applied Statistics School of Mathematics Physics and Statistics Shanghai University of Engineering Science)
  • Received : 2022.07.08
  • Accepted : 2023.07.12
  • Published : 2023.09.01
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Abstract

Zhai and Lin recently proved that if G is an n-vertex connected 𝜃(1, 2, r + 1)-free graph, then for odd r and n ⩾ 10r, or for even r and n ⩾ 7r, one has ${\rho}(G){\leq}{\sqrt{{\lfloor}{\frac{n^2}{4}}{\rfloor}}}$, 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 𝜃(1, 2, 3) or 𝜃(1, 2, 4), and we characterize all the extremal graphs. Furthermore, for n ⩾ 137, we determine the maximum number of edges in an n-vertex 𝜃(1, 2, 4)-free non-bipartite graph and characterize the unique extremal graph.

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Acknowledgement

The authors would like to express their sincere gratitude to the referee for his/her very careful reading of the paper and for insightful comments and valuable suggestions, which improved the presentation of this paper.

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