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...

In this paper we investigate the behaviour of subgraphs of sparse ε-regular bipartite graphs G = (V1 ∪ V2, E) with vanishing density d that are induced by small subsets of vertices. In particular, we show that, with overwhelming probability, a random set S ⊆ V1 of size s 1/d contains a subset S′ with |S′| ≥ (1 − ε′)|S| that induces together with V2 an ε′-regular bipartite graph of density (1 ± ...

in this paper we define the removable cycle that, if $im$ is a class of graphs, $gin im$, the cycle $c$ in $g$ is called removable if $g-e(c)in im$. the removable cycles in eulerian graphs have been studied. we characterize eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for eulerian graph to have removable cycles h...

The cyclic matching sequenceability of a simple graph G, denoted cms ( G ) , is the largest integer s for which there exists ordering edges so that every set consecutive forms matching. In this paper we consider minimum k-regular graphs. We completely determine 2-regular graphs, and give bounds k ⩾ 3.