[Paper Review] Subtrees and independent subsets in unicyclic graphs and unicyclic graphs with fixed segment sequence

Eric Ould Dadah Andriantiana, Hua Wang|arXiv (Cornell University)|Jan 22, 2020

Graph theory and applications22 references4 citations

TL;DR

This paper investigates extremal unicyclic graphs that maximize the number of subtrees and independent vertex subsets (Merrifield-Simmons index), given constraints such as fixed girth or segment sequence. It shows that the extremal structures for subtree count do not minimize the Wiener index, and those maximizing independent vertex subsets do not minimize the number of matchings—providing the first known counterexamples to the well-known negative correlation between these topological indices in unicyclic graphs with fixed segment sequences.

ABSTRACT

In the study of topological indices two negative correlations are well known: that between the number of subtrees and the Wiener index (sum of distances), and that between the Merrifield-Simmons index (number of independent vertex subsets) and the Hosoya index (number of independent edge subsets). That is, among a certain class of graphs, the extremal graphs that maximize one index usually minimize the other, and vice versa. In this paper, we first study the numbers of subtrees in unicyclic graphs and unicyclic graphs with a given girth, further confirming its opposite behavior to the Wiener index by comparing with known results. We then consider the unicyclic graphs with a given segment sequence and characterize the extremal structure with the maximum number of subtrees. Furthermore, we show that these graphs are not extremal with respect to the Wiener index. We also identify the extremal structures that maximize the number of independent vertex subsets among unicyclic graphs with a given segment sequence, and show that they are not extremal with respect to the number of independent edge subsets. These results may be the first examples where the above negative correlation failed in the extremal structures between these two pairs of indices.

Motivation & Objective

• To characterize unicyclic graphs that maximize or minimize the number of subtrees among all n-vertex unicyclic graphs.
• To investigate the relationship between the number of subtrees and the Wiener index in unicyclic graphs with fixed girth.
• To identify extremal unicyclic graphs with a given segment sequence that maximize the number of subtrees and independent vertex subsets.
• To examine whether the well-known negative correlation between topological indices (e.g., subtree count vs. Wiener index, Merrifield-Simmons index vs. Hosoya index) holds in unicyclic graphs with fixed segment sequences.
• To determine whether extremal structures for independent vertex subsets also extremize the number of matchings (Hosoya index).

Proposed method

• Define unicyclic graphs $US_n^l$ and $UP_n^l$ as extremal graphs with a cycle of length $l$ and $n-l$ pendent vertices or a pendent path, respectively.
• Use segment sequences—ordered lists of path lengths between vertices of degree ≠2—to classify unicyclic graphs and analyze extremal structures.
• Apply combinatorial lemmas on independent sets in disjoint paths, particularly using Fibonacci-based identities for the Merrifield-Simmons index.
• Prove extremality via iterative transformations: moving pendent paths to a single vertex to increase the number of independent sets or subtrees.
• Compare extremal graphs using recursive decomposition: $\sigma(G) = \sigma(G - v) + \sigma(G - [v])$, where $v$ is a vertex.
• Use induction and case analysis on segment length and number of segments to establish inequalities for $\sigma(G)$ and $n(G)$.

Experimental results

Research questions

• RQ1 Which unicyclic graphs maximize the number of subtrees among all n-vertex unicyclic graphs?
• RQ2 Does the extremal graph that maximizes the number of subtrees among unicyclic graphs with fixed girth also minimize the Wiener index?
• RQ3 For unicyclic graphs with a given segment sequence, which structure maximizes the number of subtrees?
• RQ4 Is the negative correlation between the Merrifield-Simmons index and the Hosoya index preserved in unicyclic graphs with fixed segment sequences?
• RQ5 Are the extremal graphs for the number of independent vertex subsets also extremal for the number of matchings (Hosoya index) in unicyclic graphs with fixed segment sequences?

Key findings

• The unicyclic graph $US_n^l$, formed by attaching $n-l$ pendent vertices to a single vertex of a cycle of length $l$, maximizes the number of subtrees among all n-vertex unicyclic graphs with girth $l$.
• The unicyclic graph $UP_n^l$, formed by attaching a path of length $n-l$ to a vertex of a cycle of length $l$, minimizes the number of subtrees among all n-vertex unicyclic graphs with girth $l$.
• For unicyclic graphs with a given segment sequence $(l_1, \dots, l_m)$ with $l_1 \geq 3$, the graph $U_1(l_1, \dots, l_m)$—where the cycle is a segment of length $l_1$ and all other segments form a starlike tree—maximizes the number of subtrees.
• The extremal graph that maximizes the number of subtrees among unicyclic graphs with a fixed segment sequence does not minimize the Wiener index, contradicting the typical negative correlation.
• The extremal graph that maximizes the Merrifield-Simmons index (number of independent vertex subsets) among unicyclic graphs with a fixed segment sequence is $U_2(l_1, \dots, l_m)$, where two vertices on the cycle host the longest segments.
• The extremal graph maximizing the Merrifield-Simmons index does not minimize the number of matchings (Hosoya index), providing the first known counterexample to the negative correlation between these two indices in unicyclic graphs with fixed segment sequences.