Graphs with small diameter determined by their $D$-spectra

Liu, Ruifang ; Xue, Jie

Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 1-17.

Résumé

Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},\ldots ,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of\/ $G$, where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $\lambda _{1}(D)\geq \lambda _{2}(D)\geq \nobreak \cdots \geq \lambda _{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its \hbox {$D$-spectrum} if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra.

Détail

DOI : 10.21136/CMJ.2018.0505-15

Classification : 05C50
Keywords: distance spectrum; distance characteristic polynomial; $D$-spectrum determined by its $D$-spectrum