Innovative approaches to (n,m)-rung orthopair fuzzy graphs for enhancing performance measures

Graph construction is a powerful tool for solving complex problems across various fields of computer science and computational intelligence. When dealing with uncertainty, fuzzy graph structures are preferred over crisp graph structures, as they effectively represent the inherent uncertainty within a network. To better capture vague and imprecise concepts, the (n,m)-rung orthopair fuzzy set serves as a highly effective framework, as its models provide greater flexibility than other fuzzy models when handling human judgment data. In this paper, we introduce a new framework, referred to as (n,m)-rung orthopair fuzzy graphs ((n,m)-ROFGs), by integrating the concept of graphs with (n,m)-rung orthopair fuzzy sets. The key advantage of this framework is that it meets the demands of certain applications that cannot be effectively modeled using previous approaches to handling uncertainty, as they either require equal degrees of uncertainty (such as IFSs, PFSs, and q-ROFSs) or impose restrictions on the powers of uncertainty degrees through fixed parameter values n and m( as in (2,1)-FSs and (3,2)-FSs). We start by introducing the degree and whole degree of a vertex in (n,m)-ROFGs, and illustrate their significance using road networks. Understanding the degree and whole degree of a vertex also provides insight into the characteristics of product operations on (n,m)-ROFGs. Building on this, we define the strong product and & lowast;-product as product operations on (n,m)-ROFGs. These operations become particularly useful when dealing with a large number of (n,m)-ROFGs. We also introduce the concept of vertex degree and whole degree in the strong product and & lowast;-product, and establish general theorems concerning the degree and whole degree of (n,m)-ROFGs under the proposed product operations, and provide several numerical examples to illustrate the introduced concepts. Furthermore, we explore an application of the degree and whole degree of the & lowast;-product of two (n,m)-ROFGs in multi-criteria decision-making concerning the selection of the most likely team to compete in games and using optimal performance measures. Finally, we make comparisons to highlight the shortcomings of previous techniques in modeling certain practical cases that are effectively addressed by our approach, as well as we discuss the minor limitations of the proposed method associated with excessively large values of n and m that can generally be mitigated by selecting moderate settings that better reflect typical decision-making contexts.