Graphs with a given conditional diameter that maximize the Wiener index

Abstract

The Wiener index $W(G)$ of a graph $G$ is one of the most well-known topological indices, which is defined as the sum of distances between all pairs of vertices of $G$. The diameter $D(G)$ of $G$ is the maximum distance between all pairs of vertices of $G$; the conditional diameter $D(G;s)$ is the maximum distance between all pairs of vertex subsets with cardinality $s$ of $G$. When $s=1$, the conditional diameter $D(G;s)$ is just the diameter $D(G)$. The authors in \cite{QS} characterized the graphs with the maximum Wiener index among all graphs with diameter $D(G)=n-c$, where $1\le c\le 4$. In this paper, we will characterize the graphs with the maximum Wiener index among all graphs with conditional diameter $D(G;s)=n-2s-c$ ( $-1\leq c\leq 1$), which extends partial results in \cite{QS}.

Figures (4)

• Figure 1: $G_{k,l}$ and $G_{k-1,l+1}$
• Figure 2: Tree $T^i_n$
• Figure 3: Tree $T^{i,j}_n$
• Figure 4: Tree $T^{i(2)}_n$

Theorems & Definitions (6)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3