2022/11/09 by Alraqad, Tariq A., Milovanovic, Igor Z., Saber, Hicham +3
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2211.05218
Let du be the degree of a vertex u of a graph G. The atom-bond sum-connectivity (ABS) index of a graph G is the sum of the numbers (1-2(dv+dw)-1)1/2 over all edges vw of G. This paper gives the characterization of the graph possessing the minimum ABS index in the class of all trees of a fixed number of pendent vertices; the star is the unique extremal graph in the mentioned class of graphs. The problem of determining graphs possessing the minimum ABS index in the class of all trees with n vertices and p pendent vertices is also addressed; such extremal trees have the maximum degree 3 when n≥ 3p-2≥7, and the balanced double star is the unique such extremal tree for the case p=n-2.