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The subpath number of cactus graphs

Martin Knor, Jelena Sedlar, Riste Škrekovski, Yu Yang

TL;DR

The paper investigates the subpath number $pn(G)$ for cactus graphs with fixed $n$ vertices and $k$ cycles. It develops transformations to identify extremal cacti, proving the unique maximum is the pseudo triangle chain $PTC(n,k)$ and the minimum occurs when every cycle is an end-triangle with a common-vertex structure. It shows these extremal graphs for $pn$ are not aligned with extremal graphs for the Wiener index or the subtree number, indicating $pn$ is an independent invariant. It provides closed-form expressions for $pn(PTC(n,k))$ and for the minimal end-triangle class, yielding exact characterizations and enabling precise comparisons with related indices in chemical graph theory.

Abstract

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 [5], 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, minimal cacti consist of k triangles, all sharing a common vertex, with the remaining vertices forming a tree attached to this joint vertex. 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.

The subpath number of cactus graphs

TL;DR

The paper investigates the subpath number for cactus graphs with fixed vertices and cycles. It develops transformations to identify extremal cacti, proving the unique maximum is the pseudo triangle chain and the minimum occurs when every cycle is an end-triangle with a common-vertex structure. It shows these extremal graphs for are not aligned with extremal graphs for the Wiener index or the subtree number, indicating is an independent invariant. It provides closed-form expressions for and for the minimal end-triangle class, yielding exact characterizations and enabling precise comparisons with related indices in chemical graph theory.

Abstract

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 [5], 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, minimal cacti consist of k triangles, all sharing a common vertex, with the remaining vertices forming a tree attached to this joint vertex. 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.

Paper Structure

This paper contains 4 sections, 16 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal cacti with respect to the subpath number
  4. Minimal cacti with respect to the subpath number

Key Result

Proposition 2

Let $G$ be a unicyclic graph on $n$ vertices with the cycle $C$ of length $g$. Denote by $n_{1},n_{2},\dots,n_{g}$ the number of vertices in the components that remain after removing all edges of $C$. Then,

Figures (3)

  • Figure 1: A pseudo triangle chain $\mathrm{PTC}(14,5).$
  • Figure 2: The figure shows three distinct cactus graphs from $\mathcal{C}_{10,3}.$ All these graphs minimize the subpath number in $\mathcal{C}_{10,3}.$ The leftmost graph is the pseudo friendship graph $\mathrm{PFG}(10,3)$ and only this graph minimizes Wiener index and maximizes the number of subtrees over $\mathcal{C}_{10,3}.$
  • Figure 3: The balanced saw graph $\mathrm{BSG}(14,5).$

Theorems & Definitions (16)

  • Proposition 2
  • Corollary 3
  • Corollary 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • Lemma 11
  • ...and 6 more