On the Harmonic Index and the Signless Laplacian Spectral Radius of Graphs

Hanyuan Deng ; Tomáš Vetrík ; Selvaraj Balachandran

Kragujevac Journal of Mathematics, Tome 45 (2021) no. 2, p. 299

Résumé

The harmonic index of a conected graph $G$ is defined as $H(G) =\linebreak \sum_{uv\in E(G)} \frac{2}{d(u) + d(v)}$, where $E(G)$ is the edge set of $G$, $d(u)$ and $d(v)$ are the degrees of vertices $u$ and $v$, respectively. The spectral radius of a square matrix $M$ is the maximum among the absolute values of the eigenvalues of $M$. Let $q(G)$ be the spectral radius of the signless Laplacian matrix $Q(G) = D(G) + A(G)$, where $D(G)$ is the diagonal matrix having degrees of the vertices on the main diagonal and $A(G)$ is the $(0, 1)$ adjacency matrix of $G$. The harmonic index of a graph $G$ and the spectral radius of the matrix $Q(G)$ have been extensively studied. We investigate the relationship between the harmonic index of a graph $G$ and the spectral radius of the matrix $Q(G)$. We prove that for a connected graph $G$ with $n$ vertices, we have $\frac{q(G)}{H(G)}e eft\{\begin{array}{ll} \dfrac{n^2}{2(n-1)}, \mbox{if } n\ge 6,[3mm] \dfrac{16}{5}, \mbox{if } n=5,[3mm] 3, \mbox{if } n=4, \end{array} \right.$ and the bounds are best possible.

Détail

Classification : 05C50 05C35
Keywords: harmonic index, spectral radius, eigenvalue, signless Laplacian matrix