Given a graph G, its triangular line graph is the graph T (G) with vertex set consisting of the edges of G and adjacencies between edges that are incident in G as well as being within a common triangle. Graphs with a representation as the triangular line graph of some graph G are triangular line graphs, which have been studied under many names including anti-Gallai graphs, 2-in-3 graphs, and li...

let g be a graph. the first zagreb m1(g) of graph g is defined as: m1(g) = uv(g) deg(u)2. in this paper, we prove that each even number except 4 and 8 is a first zagreb index of a caterpillar. also, we show that the fist zagreb index cannot be an odd number. moreover, we obtain the fist zagreb index of some graph operations.

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. W...

For example, the pentagon is strongly regular with parameters (v, k, λ, μ) = (5, 2, 0, 1). One easily verifies that a graph G is strongly regular with parameters (v, k, λ, μ) if and only if its complement G is strongly regular with parameters (v, v−k− 1, v− 2k+μ− 2, v−2k+λ). The line graph of the complete graph of order m, known as the triangular graph T (m), is strongly regular with parameters...

A triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, Reay and Zamfirescu showed that all 2-connected, linearly convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph D which is the linearlyconvex hull of the Star of David. We extend this result to a wider...

A (p,q) graph G is said to admit n order triangular sum labeling if its vertices can be labeled by non negative integers such that the induced edge labels obtained by the sum of the labels of end vertices are the first q n order triangular numbers. A graph G which admits n order triangular sum labeling is called n order triangular sum graph. In this paper we prove that paths, combs, stars, subd...

A graph G is k-triangular if each edge of G is in at least k triangles. It is conjectured that every 4-edge-connected 1-triangular graph admits a nowhere-zero Z3-flow. However, it has been proved that not all such graphs are Z3-connected. In this paper, we show that every 4-edge-connected 2-triangular graph is Z3-connected. The result is best possible. This result provides evidence to support t...