[Paper Review] On the Maximum Sigma Index of k-Cyclic Graphs

Akbar Ali, Abeer M. Albalahi+3 more|arXiv (Cornell University)|Jul 8, 2022

Graph theory and applications4 citations

TL;DR

This paper determines the connected $k$-cyclic graph of order $n$ that maximizes the sigma index $\sigma(G) = \sum_{uv\in E(G)}(d_u - d_v)^2$. Using degree-based optimization and graph transformation techniques, the authors prove that the graph $H_{n,k}$—formed by adding $k$ edges between a central vertex and $k$ pendent vertices of a star $S_n$—uniquely achieves the maximum sigma index for $0 \leq k \leq n-2$ and $n \geq 4$. The maximum value is $\sigma(H_{n,k}) = (n-1)(n-2)^2 - 2k(2n-5) + k^2(k-1)$.

ABSTRACT

Let $G$ be a graph with edge set $E(G)$. Denote by $d_w$ the degree of a vertex $w$ of $G$. The sigma index of $G$ is defined as $\sum_{uv\in E(G)}(d_u-d_v)^2$. A connected graph of order $n$ and size $n+k-1$ is known as a connected $k$-cyclic graph. Abdo, Dimitrov, and Gutman [Discrete Appl. Math. 250 (2018) 57-64] characterized the graphs having the greatest sigma index over the family of all connected graphs of a fixed order. The primary goal of the present note is to determine graphs possessing the greatest sigma index from the class of all connected $k$-cyclic graphs of a fixed order.

Motivation & Objective

• To determine the connected $k$-cyclic graph of fixed order $n$ that maximizes the sigma index $\sigma(G)$.
• To extend prior results on the maximum sigma index in general connected graphs to the more constrained class of $k$-cyclic graphs.
• To characterize the extremal graph structure that achieves the maximum $\sigma(G)$ within the $k$-cyclic family.
• To establish a closed-form expression for the maximum sigma index in terms of $n$ and $k$.

Proposed method

• Use of mathematical induction to prove a foundational inequality for the sigma index in graphs of size $m$.
• Application of degree-based graph transformation techniques to increase $\sigma(G)$ by relocating edges to maximize degree differences.
• Transformation of a graph $G$ with maximum degree $\Delta(G) < n-1$ into a graph $G^*$ with $\Delta(G^*) = n-1$ while increasing $\sigma(G)$.
• Decomposition of $\sigma(G)$ into contributions from the central vertex and the remaining graph $G - v$, using degree shifts.
• Application of known inequalities for the first Zagreb index $M_1(G)$ and the sigma index in star-like subgraphs.
• Derivation of the upper bound $\sigma(G) \leq (n-1)(n-2)^2 - 2k(2n-5) + k^2(k-1)$ via substitution and equality conditions from Lemmas 1 and 3.

Experimental results

Research questions

• RQ1 Which connected $k$-cyclic graph of order $n$ maximizes the sigma index $\sigma(G)$?
• RQ2 What is the exact value of the maximum sigma index for $k$-cyclic graphs of order $n$?
• RQ3 Under what structural conditions does equality hold in the upper bound for $\sigma(G)$?
• RQ4 How does the extremal graph $H_{n,k}$ differ from other $k$-cyclic graphs in terms of degree distribution?

Key findings

• The graph $H_{n,k}$, formed by connecting $k$ pendent vertices of a star $S_n$ to a central vertex, uniquely attains the maximum sigma index among all connected $k$-cyclic graphs of order $n$.
• The maximum sigma index is given by the closed-form expression $\sigma(H_{n,k}) = (n-1)(n-2)^2 - 2k(2n-5) + k^2(k-1)$ for $0 \leq k \leq n-2$ and $n \geq 4$.
• Equality in the upper bound for $\sigma(G)$ holds if and only if the subgraph induced by $G - v$ (where $v$ is the universal vertex) is isomorphic to the star $S_{k+1}$ with $n-k-2$ isolated vertices.
• The extremal graph $H_{n,k}$ is the only connected $k$-cyclic graph achieving the maximum $\sigma(G)$, establishing uniqueness of the maximizer.
• The proof relies on transforming any non-extremal $k$-cyclic graph into $H_{n,k}$ via edge-rearrangement that strictly increases $\sigma(G)$, confirming optimality.