Czechoslovak Mathematical Journal, Vol. 68, No. 1, pp. 1-17, 2018
Graphs with small diameter determined by their $D$-spectra
Ruifang Liu, Jie Xue
Received September 22, 2015. First published January 10, 2018.
Abstract: 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 $\lambda_1(D)\geq\lambda_2(D)\geq\cdots\geq\lambda_n(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its $D$-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.
Keywords: distance spectrum; distance characteristic polynomial; $D$-spectrum determined by its $D$-spectrum
Affiliations: Ruifang Liu (corresponding author), Jie Xue, School of Mathematics and Statistics, Zhengzhou University, 100 Ke Xue Road, Zhengzhou 450001, Henan, China, e-mail: rfliu@zzu.edu.cn
