Upper bounds of diminished Sombor index of graphs

The diminished Sombor index of a graph G with vertex set V ( G ) = { v 1 , v 2 , … , v n } and edge set E ( G ) is defined as D S O ( G ) = ∑ v i v j ∈ E ( G ) d v i 2 + d v j 2 d v i + d v j , in which d v i and d v j are the degrees of the adjacent vertices v i and v j . The diminished Sombor matrix of G , denoted by D S M ( G ) , is defined as the n × n matrix whose ( i , j ) − entry is d v i 2 + d v j 2 ( d v i + d v j ) − 1 if v i v j ∈ E ( G ) , and 0 otherwise. Let the eigenvalues of the diminished Sombor matrix be λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n . In this paper, we obtain two upper bounds of diminished Sombor index in terms of diminished Sombor eigenvalues of graphs, • [(i)] D S O ( G ) ≤ m λ 1 with equality holding if and only if G ≅ r K 2 ∪ ( n − 2 r ) K 1 , • [(ii)] D S O ( G ) ≤ − m λ n with equality holding if and only if G ≅ K n 1 ∪ K n 2 ∪ ⋯ ∪ K n r ∪ ( n − ∑ i = 1 r n i ) K 1 .