{
  "id": "2109.04599",
  "version": "v1",
  "published": "2021-09-10T00:26:04.000Z",
  "updated": "2021-09-10T00:26:04.000Z",
  "title": "Adjacency eigenvalues of graphs without short odd cycles",
  "authors": [
    "Shuchao Li",
    "Wanting Sun",
    "Yuantian Yu"
  ],
  "comment": "15 pages. It is accepted by Discrete Mathematics",
  "categories": [
    "math.CO"
  ],
  "abstract": "It is well known that spectral Tur\\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\\'{a}n type problem. Let $G$ be a graph and let $\\mathcal{G}$ be a set of graphs, we say $G$ is \\textit{$\\mathcal{G}$-free} if $G$ does not contain any element of $\\mathcal{G}$ as a subgraph. Denote by $\\lambda_1$ and $\\lambda_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on $\\lambda_1^{2k}+\\lambda_2^{2k}$ of $n$-vertex $\\{C_3,C_5,\\ldots,C_{2k+1}\\}$-free graphs is established, where $k$ is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most $2k+1$ in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of $n$-vertex non-bipartite graphs with odd girth at least $2k+3,$ which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-10T00:26:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C35"
    ],
    "keywords": [
      "short odd cycles",
      "adjacency eigenvalues",
      "type problem",
      "vertex non-bipartite graphs",
      "maximum spectral radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}