Invitation to the subpath number

TL;DR

The paper investigates the subpath number $pn(G)$, the total count of all subpaths including length zero, and proves its $\#P$-hardness to compute. It derives exact formulas for foundational cases (trees, unicyclic graphs, and cycles) and establishes global extremal results: $pn$ is minimized by trees and maximized by the complete graph $K_n$, with the maximum among bipartite graphs attained by $K_{\lceil n/2\rceil,\lfloor n/2\rfloor}$. It then analyzes cycle-chain graphs $G(S)$, providing a closed-form expression for $pn(G(S))$ and identifying kink chains as maximizers within a fixed cycle-length pattern, while linear and almost-linear chains minimize; special cases include ladder and hexagonal chains with explicit bounds. The work connects to chemistry through benzenoid graphs and outlines conjectures about regular and triangle-free graphs, offering several directions for future research.

Abstract

In this paper we count all the subpaths of a given graph G; including the subpaths of length zero, and we call this quantity the subpath number of G. The subpath number is related to the extensively studied number of subtrees, as it can be considered as counting subtrees with the additional requirement of maximum degree being two. We first give the explicit formula for the subpath number of trees and unicyclic graphs. We show that among connected graphs on the same number of vertices, the minimum of the subpath number is attained for any tree and the maximum for the complete graph. Further, we show that the complete bipartite graph with partite sets of almost equal size maximizes the subpath number among all bipartite graphs. The explicit formula for cycle chains, i.e. graphs in which two consecutive cycles share a single edge, is also given. This family of graphs includes the unbranched catacondensed benzenoids which implies a possible application of the result in chemistry. The paper is concluded with several directions for possible further research where several conjectures are provided.

Figures (5)

• Figure 1: All figures on the left show a path $P$ in the complete bipartite graph $K_{5,7}$ which contains $v_{b}=v_{7}.$ The figures on the right show the corresponding path $f(P)=P^{\prime}$ in $K_{6,6}$. Edges of $K_{5,7}$ and $K_{6,6}$ not contained on the path are not shown for the simplicity sake. Notice that: a) $P=u_{1}v_{2}u_{2}v_{1}u_{4}\frame{$v_7$}u_{5}v_{4}u_{3}\in\mathcal{P}_{A}$ so $P^{\prime}=u_{1}v_{2}u_{2}v_{1}u_{4}v_{3}\frame{$v_7$}v_{5}u_{5}v_{4}u_{3}\in\mathcal{P}_{A,B}^{\prime},$ b) $P=u_{1}v_{2}u_{2}v_{1}u_{4}\frame{$v_7$}u_{5}v_{4}u_{3}v_{3}\in\mathcal{P}_{A,B}$ so $P^{\prime}=u_{1}v_{2}u_{2}v_{1}u_{4}v_{6}\frame{$v_7$}v_{5}u_{5}v_{4}u_{3}v_{3}\in\mathcal{P}_{A,B}^{\prime},$ c) $P=v_{3}u_{1}v_{2}u_{2}v_{1}u_{4}\frame{$v_7$}u_{5}v_{4}u_{3}v_{6}\in\mathcal{P}_{B}$ so $P^{\prime}=u_{1}v_{3}u_{2}v_{2}u_{4}v_{1}\frame{$v_7$}v_{4}u_{5}v_{6}u_{3}\in\mathcal{P}_{A}.$
• Figure 2: Graph $G(S)$ with $S=(3,2,1,1,4;3,1,3,2,4)$.
• Figure 3: The figure shows the only $4$-chain with $k=5.$
• Figure 4: The figure shows hexagonal chains: a) the linear chain $L_{5},$ b) the zig-zag chain $Z_{5}$ and c) the helicene $H_{5}.$
• Figure 5: The graphs: a) $L_{18},$ b) $L_{20}.$

Theorems & Definitions (16)

• Proposition 1
• Proposition 3
• Corollary 4
• Corollary 5
• Proposition 6
• Corollary 7
• Proposition 8
• Lemma 9
• Theorem 10
• Theorem 11