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<p>The subpath number of a graph <i>G</i> is defined as the total number of subpaths in <i>G</i>, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper Knor et al. (Knor M, Sedlar J, Škrekovski R, et al (2026) Invitation to the subpath number[J]. Appl Math Comput 509:129646), where we investigated the subpath number and identified extremal graphs within the classes of trees, unicyclic graphs, bipartite graphs, and cycle chains. Here, we focus on the subpath number of cactus graphs and characterize all maximal and minimal cacti with <i>n</i> vertices and <i>k</i> cycles. We prove that maximal cacti are cycle chains in which all interior cycles are triangles, while the two end-cycles differ in length by at most one. In contrast, the minimal cacti consist of <i>k</i> cycles, all of which are end-triangles, with the subgraph induced by the remaining vertices forming a forest. By comparing extremal cacti with respect to the subpath number to those that are extremal for the subtree number and the Wiener index, we demonstrate that the subpath number does not correlate with either of these quantities, as their corresponding extremal graphs differ.</p>

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The subpath number of cactus graphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月15日

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  • Martin Knor,
  • Jelena Sedlar,
  • Riste Škrekovski,
  • Yu Yang

摘要

The subpath number of a graph G is defined as the total number of subpaths in G, and it is closely related to the number of subtrees, a well-studied topic in graph theory. This paper is a continuation of our previous paper Knor et al. (Knor M, Sedlar J, Škrekovski R, et al (2026) Invitation to the subpath number[J]. Appl Math Comput 509:129646), where we investigated the subpath number and identified extremal graphs within the classes of trees, unicyclic graphs, bipartite graphs, and cycle chains. Here, we focus on the subpath number of cactus graphs and characterize all maximal and minimal cacti with n vertices and k cycles. We prove that maximal cacti are cycle chains in which all interior cycles are triangles, while the two end-cycles differ in length by at most one. In contrast, the minimal cacti consist of k cycles, all of which are end-triangles, with the subgraph induced by the remaining vertices forming a forest. By comparing extremal cacti with respect to the subpath number to those that are extremal for the subtree number and the Wiener index, we demonstrate that the subpath number does not correlate with either of these quantities, as their corresponding extremal graphs differ.