On extremal Sombor index of bipartite graphs with given order and diameter

Abstract

We consider the extremal value of the Sombor index in the class of bipartite graphs of a given order $n$ and diameter $d$. Using a layered perspective on bipartite graphs, we first derive structural lemmas and an exact formula for the Sombor index $SO(G)$ in terms of the vertex partition sizes at successive distances from a fixed vertex. This framework allows us to transform the problem into a constrained optimization, which we solve by applying concavity and convexity arguments. As a main result, we prove a comprehensive characterization of the graphs that maximize $SO(G)$ for each fixed $n$ and $d$. In particular, our results show that for $d=3$ the unique extremal graph has one vertex in each of the distance-$0$ and distance-$3$ partitions and splits the remaining $n-2$ vertices as evenly as possible between the distance-$1$ and distance-$2$ partitions. For any $d\ge 4$, the graph with maximum Sombor index is obtained by placing a single vertex in each outer distance partition while distributing the other $n-(d+1)$ vertices between two inner partitions (one of which receives $\lceil (n-d+1)/2\rceil$ vertices and the other $\lfloor (n-d+1)/2\rfloor$). These findings fill the gap in the extremal graph theory of the Sombor index under diameter constraints, extending the known case of unconstrained bipartite graphs (attaining their maximum $SO$ at $d=2$) to arbitrary prescribed diameters.