Arithmetic–Geometric and Geometric–Arithmetic Index on Fuzzy Graphs and their Real Life Applications

In this article, the notion of arithmetic–geometric index (AGI) and geometric-arithmetic index (GAI) of a fuzzy graph (FG) are introduced and some important results regarding these indices are investigated. Some bounds of AGI are established in the form of some theorems. The AGI of the disjoint union of two FGs and direct product of two FGs are investigated. Relationship between AGI and GAI is established. At the same time, a condition is proposed under which these two indices are equal. Inequalities regarding the sum, product and ratio of these indices are established through several theorems. Also, the concepts of exponential AGI and exponential GAI are initiated and some important bounds of these indices are provided in the form of some theorems. Moreover, different types of nodes in a FG are introduced in terms of average AGI and some crucial results are established in this connection. Lastly, applications of AGI are described in Telecommunication and Hospital management network. These show how important the AGI is to measure imbalance in a system and to resolve these imbalances accordingly. Significance of the work: The paper contributes to FG theory through the introduction of AGI and GAI and their novel properties. This paper provides several bounds, inequalities, and relationships among AGI and GAI. Also, the conditions when these two indices coincide are found. An exponential formulation of AGI and GAI is provided to further deepen the analysis. The classification of nodes using average AGI helps in comprehending FGs better. The applications in telecommunication and hospital networks where there is an imbalance, are considered.