[Paper Review] Cacti with Extremal PI Index

Chunxiang Wang, Shaohui Wang+1 more|arXiv (Cornell University)|Mar 1, 2016

Medicinal Plants and Neuroprotection16 references21 citations

TL;DR

This paper determines the cacti graphs with the maximum and minimum vertex PI indices among all cacti of order $n$ and with $k$ pendant vertices. Using structural graph theory and extremal graph analysis, it establishes sharp upper and lower bounds for the vertex PI index, showing that extremal cacti are characterized by specific cycle structures (even cycles for maximum, triangles for minimum) and optimal edge distributions, extending prior results on topological indices in chemical graph theory.

ABSTRACT

The vertex PI index $PI(G) = \sum_{xy \in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the extremal graphs with the largest and smallest vertex PI indices among all the cacti. As a consequence, we obtain the sharp bounds with corresponding extremal cacti and extend a known result.

Motivation & Objective

• To identify the cacti with the largest and smallest vertex PI indices among all cacti of order $n$ and $k$ pendant vertices.
• To extend previous results on topological indices by excluding constraints on the number of edges and cliques.
• To establish sharp upper and lower bounds for the vertex PI index in cacti, independent of edge or clique count.
• To characterize the structural properties of extremal cacti achieving these bounds, particularly regarding cycle types and cut edges.

Proposed method

• Analyzes the vertex PI index $PI(G) = \sum_{xy \in E(G)} [n_{xy}(x) + n_{xy}(y)]$, where $n_{xy}(x)$ counts vertices closer to $x$ than $y$.
• Applies structural lemmas to restrict cycle types: even cycles for maximal PI, triangles for minimal PI, under extremality constraints.
• Uses edge-replacement operations to transform graphs and prove strict decrease/increase in PI index, establishing optimality.
• Applies induction and case analysis on cycle length and path structure to derive extremal configurations.
• Employs combinatorial counting to compute $PI(G)$ based on cycle length and number of cut edges.
• Derives closed-form expressions for extremal PI values using floor functions: $\lfloor \frac{n-k-1}{3} \rfloor$ for max, $\lfloor \frac{n-k-1}{2} \rfloor$ for min.

Experimental results

Research questions

• RQ1 What is the maximum vertex PI index among all cacti of order $n$ with $k$ pendant vertices?
• RQ2 What is the minimum vertex PI index among all cacti of order $n$ with $k$ pendant vertices?
• RQ3 What structural properties characterize the cacti that achieve the maximum and minimum vertex PI indices?
• RQ4 How do cycle lengths (e.g., 4, 6, or 3) influence the extremal values of the vertex PI index?
• RQ5 Can the extremal cacti be uniquely characterized, and what are the sharp bounds for the vertex PI index in terms of $n$ and $k$?

Key findings

• The maximum vertex PI index among cacti with $n$ vertices and $k$ pendant vertices is $(n-1 + \lfloor \frac{n-k-1}{3} \rfloor)(n-2)$, achieved when all cycles are of length 4 or 6 and the graph has a central vertex with $k$ pendent edges and a path of length at most 2.
• The minimum vertex PI index is $(n-1)(n-2) - 2\lfloor \frac{n-k-1}{2} \rfloor$, achieved when all cycles are triangles and the graph has at most $k+1$ cut edges.
• Extremal cacti for maximum PI index are characterized by having all cycles of even length (4 or 6), with at most one cycle of length 6 and the rest of length 4.
• Extremal cacti for minimum PI index are characterized by having all cycles of length 3, with a central vertex connected to $k-1$ pendent edges and a path of length at most 2.
• The number of cycles in the extremal graphs is $\lfloor \frac{n-k-1}{3} \rfloor$ for maximum PI and $\lfloor \frac{n-k-1}{2} \rfloor$ for minimum PI.
• The extremal cacti are not unique, but all graphs satisfying the structural conditions in Theorems 1 and 2 achieve the respective sharp bounds.