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The main objective of this research article is to classify different types of m-polar fuzzy edges in an m-polar fuzzy graph by using the strength of connectedness between pairs of vertices. The identification of types of m-polar fuzzy edges, including α-strong m-polar fuzzy edges, β-strong m-polar fuzzy edges and δ-weak m-polar fuzzy edges proved to be very useful to completely determine the basic structure of m-polar fuzzy graph. We analyze types of m-polar fuzzy edges in strongest m-polar fuzzy path and m-polar fuzzy cycle. Further, we define various terms, including m-polar fuzzy cut-vertex, m-polar fuzzy bridge, strength reducing set of vertices and strength reducing set of edges. We highlight the difference between edge disjoint m-polar fuzzy path and internally disjoint m-polar fuzzy path from one vertex to another vertex in an m-polar fuzzy graph. We define strong size of an m-polar fuzzy graph. We then present the most celebrated result due to Karl Menger for m-polar fuzzy graphs and illustrate the vertex version of Menger’s theorem to find out the strongest m-polar fuzzy paths between affected and non-affected cities of a country due to an earthquake. Moreover, we discuss an application of types of m-polar fuzzy edges to determine traffic-accidental zones in a road network. Finally, a comparative analysis of our research work with existing techniques is presented to prove its applicability and effectiveness.

<abstract><p>In this research study, we first define the strong degree of a vertex in an $ m $-polar fuzzy graph. Then we present various useful properties and prove some results concerning this new concept, in the case of complete $ m $-polar fuzzy graphs. Further, we introduce the concept of $ m $-polar fuzzy strength sequence of vertices, and we also investigate it in the particular instance of complete $ m $-polar fuzzy graphs. We discuss connectivity parameters in $ m $-polar fuzzy graphs with precise examples, and we investigate the $ m $-polar fuzzy analogue of Whitney's theorem. Furthermore, we present a clustering method for vertices in an $ m $-polar fuzzy graph based on the strength of connectedness between pairs of vertices. In order to formulate this method, we introduce terminologies such as $ \epsilon_A $-reachable vertices in $ m $-polar fuzzy graphs, $ \epsilon_A $-connected $ m $-polar fuzzy graphs, or $ \epsilon_A $-connected $ m $-polar fuzzy subgraphs (in case the $ m $-polar fuzzy graph itself is not $ \epsilon_A $-connected). Moreover, we discuss an application for clustering different companies in consideration of their multi-polar uncertain information. We then provide an algorithm to clearly understand the clustering methodology that we use in our application. Finally, we present a comparative analysis of our research work with existing techniques to prove its applicability and effectiveness.</p></abstract>

The Zagreb indices are very popular topological indices in mathematical chemistry and attracted a lot of attention in recent years. The first and second Zagreb indices of a graph G = (V, E) are defined as M 1 (G) = vi∈V d 2 ı and M 2 (G) = υi∼υȷ (didȷ), where di denotes the degree of a vertex υi and υi ∼ υȷ represents the adjacency of vertices υi and υȷ in G. It has been conjectured that M 1 /n ≤ M 2 /m holds for a connected graph G with n = |V | and m = |E|. Later, it is proved that this inequality holds for some classes of graphs but does not hold in general. This inequality is proved to be true for graphs with di ∈, where h ≥ z(z − 1)/2. In this paper, we prove that the graphs satisfy the inequality if the sequences (di) and (Si) have the similar monotonicity, where Si = υȷ∈N (υi) dȷ and N (υi) = {υȷ ∈ V |υi ∼ υȷ}. As a consequence, we present an infinite family of connected graphs with di ∈ [1, ∞), for which the inequality holds. Moreover, we establish the relations between M 1 /n and M 2 /m in case of general graphs.

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