Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characte...

the main aim of this paper is to extend the recently developed methods for calculating the buckling loads of planar symmetric frames to include the effect of semi-rigidity of the joints. this is achieved by decomposing a symmetric model into two submodels and then healing them in such a manner that the ::::union:::: of the eigenvalues of the healed submodels result in the eigenvalues of the ent...

Complex real-world networks such as the web graph are often modelled as directed graphs evolving over time, where new vertices are joined to a constant m number of existing vertices of prescribed type. We consider a certain on-line random construction of a countably infinite graph with out-degree m, and show that with probability 1 the construction gives rise to a unique isomorphism type. We sh...

a vertex irregular total k-labeling of a graph g with vertex set v and edge set e is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. the total vertex irregularity strength of g, denoted by tvs(g)is the minimum value of the largest label k over all such irregular assignment. in this paper, we study the to...

In this note, we obtain the expressions for multiplicative Zagreb indices and coindices of derived graphs such as a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph and paraline graph.

Let R be a commutative ring with identity and )) ( ( R T Γ its total graph. The subject of this article is the investigation of the properties of the corresponding line graph ))). ( ( ( R T L Γ In particular, we determine the girth and clique number of ))). ( ( ( R T L Γ In addition to that, we find the condition for ))) ( ( ( R T L Γ to be Eulerian.

The k-th semi-total point graph R(G) of a graph G, is a graph obtained from G by adding k vertices corresponding to each edge and connecting them to the endpoints of the edge considered. In this paper, we obtain formulas for the resistance distance and Kirchhoff index of R(G).