Analytical Computation of the Total Kulli-Basava Polynomial for Wheel, Gear and Helm Graphs

Abstract

In this paper, we study the Total Kulli Basava Polynomial (TKBP), which is defined using the absolute differences of the sums of degrees of adjacent vertices. We first establish general properties of the TKBP for regular graphs, demon strating that the polynomial takes on a constant value in such cases. Additionally, we derive explicit closed-form expressions for wheel graphs, gear graphs, and helm graphs. Our approach relies on a systematic classification of vertices based on their edge-degree sums, along with a careful enumeration of contributions from each edge.We also examine the properties of these polynomials in relation to the graphs men
tioned, highlighting patterns and structural characteristics that arise from their construction