Wiener index and traceable graphs (Q2872004)

The Wiener number \(W(G)\) of a connected graph \(G\) is a well-known distance-based chemical index, which is defined as the sum of distances between all pairs of vertices in \(G\). A graph is called traceable if it has a path including all vertices. Applying Chvátal's degree condition for a graph to be traceable [\textit{V. Chvátal}, J. Comb. Theory, Ser. B 12, 163--168 (1972; Zbl 0213.50803)], this short paper obtained a new sufficient condition in terms of the Wiener number: For a simple connected graph of order \(n\geq 4\), if \(W(G)\leq (n+5)(n-2)/2\), then \(G\) is traceable unless \(G\) is isomorphic to \(K_1+(K_{n-3}\cup 2K_1)\) or \(K_2+(3K_1\cup K_2)\) or \(K_4+6K_1\).