In different fields in our life, like physics, coding theory and computer science, graph labeling dramas an vital role and appears in many applications. A labeling of a graph is a map which assign each element in with a positive integer number. An edge irregular total -labeling is a function such that where and are weights for any two distinct edges. In this case, has total edge irregularity strength (TEIS) if is minimum. In our paper, we defined a new type of graphs called a triple star snake graph and star snake graph . Also, we investigated TEIS for a triple star snake graph . After that, we generalized the results for star snake graph .

A total $m$-labelling $\alpha$ on graph $\Gamma$ is called an edge irregular total $m$-labelling if for any two different edges of $\Gamma$, their weights respect to $\alpha$ are distinct, where the weight of any edge is defined as the sum of its label and the labels of its end vertices. We determine the minimum $m$ such that $\Gamma$ can be labelled by an edge irregular total $m$-labelling whenever $\Gamma$ is an odd staircase graph or an even staircase graph.

We investigate new graph characteristics namely total (vertex, edge) face irregularity strength of gen- eralized plane grid graphs Gmn under k-labeling Phi of type (Alpha, Beta, Gamma). The minimum integer k for which a vertex-edge labelled graph has distinct face weights is called the total face irregularity strength of the graph and is denoted by tfs(Gmn). In this article, the graphs G = (V;E; F) under consideration are simple, finite, undirected and planar. We will estimate the exact tight lower bounds for the total face irregularity strength of some families of generalized plane grid graphs.

In general, isolators and dampers used in seismically isolated buildings are designed to be isotropic in any horizontal direction. However, in the case of buildings with plan irregularities, their nonlinear responses depend on the direction of seismic loading. To discuss the influence of the angle of seismic incidence (ASI) on the nonlinear response of irregular building structures, it is important to define the angle of the critical axis of the horizontal ground motion. One possible choice is the “principal axis of ground motion” proposed by Arias (1970). However, because this principal axis is independent of the natural period of a structure, it could be complicated to use for seismically isolated structures with long natural periods. In this study, the influence of the ASI of long-period pulse-like seismic input on an irregular base-isolated building is investigated. First, the angle of the principal axis of ground motion is defined in terms of the cumulative energy input. Then, a nonlinear time-history analysis of a five-story irregular base-isolated building is performed using 10 long-period pulse-like ground motion records considering various ASIs. The results show that, compared with the principal axis of ground motion proposed by Arias, defining the principal axis of ground motion in terms of the cumulative energy input is more suitable for discussions concerning the influence of the ASI on the response of an irregular base-isolated building.