Isomorphisms of generalized Cayley graphs

Let G be a simple connected graph and $x,y\in V(G).$ Let $H_G(x,y)$ be the expected hitting time from x to y in G and $\phi (G)$ be the hitting time of G, where $\phi (G)=\max \{H_G(x,y):x,y\in V(G)\}.$ A tricyclic graph is a simple connected graph that the edge number equals the vertex number plus two. Let $\mathcal{T}(n)$ be the set of all n-vertex tricyclic graphs. In this article, we will determine the extremal graphs for hitting times among all n-vertex tricyclic graphs. Moreover, if $G\in \mathcal{T}(n),$ then we will obtain sharp upper and lower bounds for $\phi (G).$

We explore the extremal problems of the hitting time of unicyclic graphs on n vertices with a given diameter. Let $ \mathrm{H_G} $(u, v) be the expected hitting time from vertex u to vertex v on a simple graph G. Let φ(G) = $ \mathrm{max_{u}{v}{∈}{V}{(G)}} $, $ \mathrm{H_G} $ (u,v) be the hitting time of G. In this paper, we obtain the upper bound for the hitting time of unicyclic graphs with a given diameter, and the extremal graph that attached the value is determined.