关键词: 距离谱 ,  距离整谱图 ,  强和 ,  强积

Abstract: For two connected graphs $G$ and $H$, the strong sum $G\oplus H$ is the graph with vertex set $V(G)\times V(H)$ and edge set $\{(u,v)(u',v')\mid uu'\in E(G),v=v'\}\cup\{(u,v)(u',v')\mid uu'\in E(G),vv'\in E(H)\}$, and the strong product $G\otimes H$ is the graph with vertex set $V(G)\times V(H)$ and edge set $\{(u,v)(u',v')\mid uu'\in E(G),v=v'\}\cup\{(u,v)(u',v')\mid uu'\in E(G),vv'\in E(H)\}\cup\{(u,v)(u',v')\mid u=u',vv'\in E(H)\}$. In this paper we completely obtain the distances in the $G\oplus H$ and $G\otimes H$ when $H$ has diameter less than $3$. Furthermore, we get the distance spectra of $G\oplus H$ and $G\otimes H$ when $G$ and $H$ satisfy some conditions. As applications, some distance integral graphs generated by strong sum and strong product are obtained. Especially, we get a new infinite class of distance integral graphs generated by strong product.

Key words: Distance spectrum ,  Distance integral graph ,  , Strong sum ,  , Strong product