Graph theory has provided a very useful tool, called topological indices which are a number obtained from the graph $G$ with the property that every graph $H$ isomorphic to $G$, value of a topological index must be same for both $G$ and $H$. In this article, we present exact expressions for some topological indices of k-th subdivision graph and semi total point graphs respectively, which are a ...

Let H be a fixed graph. What can be said about graphs G that have no subgraph isomorphic to a subdivision of H? Grohe and Marx proved that such graphs G satisfy a certain structure theorem that is not satisfied by graphs that contain a subdivision of a (larger) graph H1. Dvořák found a clever strengthening—his structure is not satisfied by graphs that contain a subdivision of a graph H2, where ...

LetH be a multigraph, possibly containing loops. AnH-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n0(H) isolated vertices and n1(H) vertices of degree one. In [10] it was shown that if G is a simple graph of order n containing an H-subdivision H and δ(G) ≥ n+m−k+n1(H)+2n0(H) 2 , then G c...

Mader conjectured that every C4-free graph has a subdivision of a clique of order linear in its average degree. We show that every C6-free graph has such a subdivision of a large clique. We also prove the dense case of Mader’s conjecture in a stronger sense, i.e. for every c, there is a c′ such that every C4-free graph with average degree cn 1/2 has a subdivision of a clique K` with ` = bc′n1/2...

Neutrosophic theory has many applications in graph theory, interval valued neutrosophic graph (IVNG) is the generalization of fuzzy graph, intuitionistic fuzzy graph and single valued neutrosophic graph. In this paper, we introduced some types of IVNGs, which are subdivision IVNGs, middle IVNGs, total IVNGs and interval valued neutrosophic line graphs (IVNLGs), also discussed the isomorphism, c...

The aim of this work is to investigate the F-index and F-coindex of the line graphs of the cycle graphs, star graphs, tadpole graphs, wheel graphs and ladder graphs using the subdivision concepts. F-index of the line graph of subdivision graph of square grid graph, 2D-lattice, nanotube and nanotorus of TUC4C8[p, q] are also investigated here.

This is best possible in the sense that we cannot additionally require the entire bipartite subgraph of G between S and T to have large minimum degree. Applying the result of Mader that every graph of sufficiently large average degree contains a subdivision of a given graph H to the graph G[S] obtained by the above theorem, it immediately follows that every highly connected graph G contains a “...

A k-stack layout (respectively, k-queue layout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A k-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The sta...