Resistance Distance and Spanning Trees of Multiple Extended Complete Split-Like Graph

The article explores the resistance distance and spanning trees of multiple extended complete split-like graphs, building upon foundational concepts in graph theory. It begins by defining key graph operations such as the join and Cartesian product of graphs, and introduces the concept of resistance distance, an important invariant in graph theory. The study delves into various types of join and product graphs, highlighting previous research and setting the stage for the main results. The core of the article presents detailed theorems and proofs for calculating resistance distances and counting spanning trees in multiple extended complete split-like graphs. It employs principles such as series and parallel resistance, elimination, and substitution to derive these results. The article also includes practical applications and computational methods, demonstrating the efficiency of the proposed theorems through MATLAB simulations. The detailed examples and comparisons with existing methods underscore the significance of the findings, making this article a crucial read for anyone interested in advanced graph theory and its applications.