On a Conjecture Concerning the Complementary Second Zagreb Index

TL;DR

The paper tackles the problem of maximizing the complementary second Zagreb index $cM_2$ over connected graphs of order $n$ and tests the conjecture that the extremal graph is the join $K_k+\overline{K}_{n-k}$ with $k<\lceil n/2\rceil$. It develops structural results, showing the extremal graph must satisfy $\Delta=n-1$ and that minimum-degree vertices form an independent set, and derives a sharp bound on the number of maximum-degree vertices $|M(G)|$. It extends these results to classes of bidegreed and tridegreed graphs, using comparisons to $K_t+\overline{K}_{n-t}$ and a lemma for the tridegreed case, and supplements with computational enumeration of admissible $k$ values for $5\le n\le149$, noting the resulting sequence is not present in OEIS. Overall, the work strengthens empirical support for the conjecture and clarifies the extremal landscape, while highlighting remaining challenges in determining the exact $k$ as a function of $n$.

Abstract

The complementary second Zagreb index of a graph $G$ is defined as $cM_2(G)=\sum_{uv\in E(G)}|(d_u(G))^2-(d_v(G))^2|$, where $d_u(G)$ denotes the degree of a vertex $u$ in $G$ and $E(G)$ represents the edge set of $G$. Let $G^*$ be a graph having the maximum value of $cM_2$ among all connected graphs of order $n$. Furtula and Oz [MATCH Commun. Math. Comput. Chem. 93 (2025) 247--263] conjectured that $G^*$ is the join $K_k+\overline{K}_{n-k}$ of the complete graph $K_k$ of order $k$ and the complement $\overline{K}_{n-k}$ of the complete graph $K_{n-k}$ such that the inequality $k<\lceil n/2 \rceil$ holds. We prove that (i) the maximum degree of $G^*$ is $n-1$ and (ii) no two vertices of minimum degree in $G^*$ are adjacent; both of these results support the aforementioned conjecture. We also prove that the number of vertices of maximum degree in $G^*$, say $k$, is at most $-\frac{2}{3}n+\frac{3}{2}+\frac{1}{6}\sqrt{52n^2-132n+81}$, which implies that $k<5352n/10000$. Furthermore, we establish results that support the conjecture under consideration for certain bidegreed and tridegreed graphs. In the aforesaid paper, it was also mentioned that determining the $k$ as a function of the $n$ is far from being an easy task; we obtain the values of $k$ for $5\le n\le 149$ in the case of certain bidegreed graphs by using computer software and found that the resulting sequence of the values of $k$ does not exist in "The On-Line Encyclopedia of Integer Sequences" (an online database of integer sequences).

Figures (1)

• Figure 1: The graph $K_k+\overline{K}_{n-k}$.

Theorems & Definitions (15)

• Conjecture 1
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Corollary 1
• Proposition 5
• proof