A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...

A vertex-coloured graph G is said to be rainbow vertex-connected, if every two vertices of are connected by a path whose internal have distinct colours. The vertex-connection number G, denoted rvc(G), the smallest colours that needed make vertex-connected. This study focuses on deriving formulas for vertex connectivity simple ladder and roach graph.

We present the characteristic polynomial for transition matrix of a vertex-face walk on graph, and obtain its spectra. Furthermore, we express 2-dimensional torus by using adjacency matrix, As an application, define new walk-type zeta function with respect to two-dimensional torus, explicit formula.