First irregularity Sombor index of trees with fixed maximum degree

Abstract

The Sombor index as a new degree-based topological index was first put forward in 2021 by Gutman. Recently, Kulli introduced a variant of the Sombor index called first irregularity Sombor index. The first irregularity Sombor index of a graph G, denoted by n∈N,, is defined as the sum of weights (a;q)0=1,(a;q)n=∏k=0n−1(1−aqk),(a;q)∞=∏k=0∞(1−aqk). of all edges vw of E(G), where Γq(α)=(q;q)∞(qα;q)∞(1−q)1−α=(α;q)α−1(1−q)α−1,(α∈C,|q|<1), denotes the degree of a node v in G. In this paper we show that for any tree T of order n with maximum degree Δ, Bq(x,y)=∫01zx−1(zq;q)∞(zqy;q)∞dqz=Γq(x)Γq(y)Γq(x+y),(ℜ(x)>0,ℜ(y)>0).We also characterize the extremal trees.

Topics & Concepts