[Paper Review] Zeroth-order general Randic index of $k$-generalized quasi trees

Muhammad Kamran Jamil, Ioan Tomescu|arXiv (Cornell University)|Jan 11, 2018

Graph theory and applications8 references3 citations

TL;DR

This paper characterizes the extremal $k$-generalized quasi-trees that minimize and maximize the zeroth-order general Randić index $^0R_eta(G) = \sum_{v \in V(G)} d(v)^\alpha$ for $\alpha \neq 0$. Using graph-theoretic optimization and degree sequence analysis, it establishes tight bounds for $\alpha < 0$, $0 < \alpha < 1$, $\alpha = 1$, and $\alpha > 1$, showing that extremal graphs are either complete joins with stars or have nearly regular degree distributions depending on $\alpha$. The results extend prior work on Zagreb and Randić indices to a broader class of graphs.

ABSTRACT

For a simple graph $G(V,E)$, the zeroth-order general Randi\' c index is defined as $^0R_α(G)=\sum_{v\in V(G)}d(v)^α$, where $d(v)$ is the degree of the vertex $v$ and $α e0$ is a real number. The $k$-generalized quasi-tree is a connected graph $G$ with a subset $V_k\subset V(G)$, where $|V_k|=k$ such that $G-V_k$ is a tree, but for any subset $V_{k-1}\subset V(G)$ with cardinality $k-1$, $G-V_{k-1}$ is not a tree. In this paper, we characterize the extremal $k$-generalized quasi trees with the minimum and maximum values of the zeroth-order general Randi\' c index for $α eq 0$.

Motivation & Objective

• To determine the $k$-generalized quasi-trees of order $n$ that minimize and maximize the zeroth-order general Randić index $^0R_\alpha(G)$ for $\alpha \neq 0$.
• To extend prior results on extremal indices in trees and quasi-trees to the broader class of $k$-generalized quasi-trees.
• To establish sharp upper and lower bounds for $^0R_\alpha(G)$ based on the value of $\alpha$, considering structural constraints of $k$-quasi vertices.
• To characterize the exact graph structures achieving extremal index values under different $\alpha$ regimes.
• To generalize findings from $k=1$ (studied by Qiao) to arbitrary $k \geq 1$.

Proposed method

• Utilizes the zeroth-order general Randić index $^0R_\alpha(G) = \sum_{v \in V(G)} d(v)^\alpha$, where $\alpha \neq 0$ is a real parameter.
• Applies degree sequence optimization via Lemmas 1–7, particularly leveraging convexity/concavity of $x^\alpha$ to compare index values under degree redistribution.
• Employs graph operations such as edge addition ($G + uv$) and vertex degree adjustment ($G' = G + uw - vw$) to prove extremality under different $\alpha$ regimes.
• Uses the join operation $K_k + T_{n-k}$ to construct candidate extremal graphs, especially for $\alpha > 1$ and $\alpha = 1$.
• Applies structural constraints: $k$-quasi vertices induce a complete subgraph in extremal graphs for $\alpha > 1$, and have degree 2 in minimal graphs for $\alpha < 0$.
• Applies extremal graph theory results (e.g., extremal trees for $\alpha < 0$ or $\alpha > 1$) to the $k$-generalized quasi-tree framework.

Experimental results

Research questions

• RQ1 What is the minimum and maximum value of the zeroth-order general Randić index $^0R_\alpha(G)$ over all $k$-generalized quasi-trees of order $n$?
• RQ2 How do the extremal graphs depend on the exponent $\alpha$ in $^0R_\alpha(G) = \sum d(v)^\alpha$?
• RQ3 For which graph structures is $^0R_\alpha(G)$ minimized or maximized when $\alpha < 0$, $0 < \alpha < 1$, $\alpha = 1$, and $\alpha > 1$?
• RQ4 Can the extremal graphs be explicitly characterized in terms of their degree sequences and connectivity patterns?
• RQ5 How do the results generalize prior findings for $k=1$ and for first and second Zagreb indices?

Key findings

• For $\alpha > 1$, the maximum of $^0R_\alpha(G)$ is achieved if and only if $G = K_k + S_{n-k}$, with the upper bound $ (k+1)(n-1)^\alpha + (n-k-1)(k+1)^\alpha $.
• For $\alpha > 1$, the minimum of $^0R_\alpha(G)$ is $ (n-2k+2)2^\alpha + (2k-2)3^\alpha $, achieved when $G$ has $n-2k+2$ vertices of degree 2 and $2k-2$ vertices of degree 3.
• For $\alpha = 1$, the minimum value is $2(n+k-1)$, achieved when all $k$-quasi vertices have degree 2, and the maximum is $2n(k+1) - k(k+3) - 2$, achieved when $G = K_k + T_{n-k}$ with $T_{n-k}$ arbitrary.
• For $0 < \alpha < 1$, the maximum is $k(n-1)^\alpha + 2(k+1)^\alpha + (n-k-2)(k+2)^\alpha$, achieved if and only if $G = K_k + P_{n-k}$.
• For $k=1$ and $0 < \alpha < 1$, the minimum is $ (n-1)^\alpha + 2^{\alpha+1} + n - 3 $, achieved when $G = K_1 \bullet_{u,v} S_{n-1}$ with $u$ center and $v$ pendant vertex.
• For $k \geq 2$ and $0 < \alpha < 1$, the minimum is $ (n-2)^\alpha + k \cdot 2^\alpha + (k+2)^\alpha + n - k - 2 $, achieved when $G = \overline{K_k} \bullet_{u,v} S_{n-k-2,2}(u,v)$ with $u$ and $v$ of degrees $n-k-2$ and $2$ respectively.