Revista de la
Unión Matemática Argentina
On the sum of the eigenvalues of the distance Laplacian matrix of graphs with diameter three and four
Ummer Mushtaq, Shariefuddin Pirzada, and Saleem Khan

Volume 69, no. 1 (2026), pp. 193–202    

Published online (final version): February 3, 2026

https://doi.org/10.33044/revuma.5147

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Abstract

We study an inequality proposed by Zhou et al. (2025) relating the distance Laplacian eigenvalues of a connected graph to its Wiener index. For a connected graph $G$ on $n$ vertices, let $U_r(G)$ denote the sum of the $r$ largest distance Laplacian eigenvalues, and let $W(G)$ be the Wiener index of $G$. Zhou et al. conjectured that, for all $r=2,\dots,n$, one has $U_r(G) \le W(G) + \binom{r+2}{3}$. We prove this inequality for several families of graphs. In particular, for $n\ge 95$ we verify it for all graphs in $\Gamma_n$, that is, graphs of order $n$ and diameter 3 that contain a spanning tree of diameter 3; as a consequence, the conjecture holds for all trees of diameter 3. Moreover, we show that if $G$ has maximum degree $n-2$, then the inequality holds for all $1\le r\le n$. Finally, we prove that sun graphs and partial sun-type graphs of diameter 4 also satisfy the inequality for all $1\le r\le n$.

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