The computation of resistance distance and the Kirchhoff index is a fundamental problem in graph theory. These metrics provide insights into network behavior and are crucial for analyzing network robustness and optimizing communication systems. The resistance distance \(r_G(a, b)\) between two vertices a and b in a graph G is the effective resistance in an equivalent electrical network where each edge represents a unit resistor. The Kirchhoff index Kf(G) is the sum of all resistance distances between pairs of vertices. This paper focuses on the cyclic octahedral graph COS(N), formed by arranging N octahedra in a cyclic sequence. Using computational techniques, including series–parallel rules, \(\Delta -Y\) transformations, the reduction principle, and the principle of elimination, we derive the resistance distances between vertex pairs and an exact formula for the Kirchhoff index of COS(N).
