@article{Wu2023, 
author = {Fan Wu and Xinhui An and Baoyindureng Wu},
title = {Sombor indices of cacti},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {1},
pages = {1550-1565},
keywords = {Sombor index, cactus, extreme value},
url = {https://www.sciopen.com/article/10.3934/math.2023078},
doi = {10.3934/math.2023078},
abstract = {For a graph    G, the Sombor index    S  O  (  G  ) of    G is defined as     S  O  (  G  )  =      ∑          u      v      ∈      E      (      G      )                  d              G              (    u          )              2              +          d              G              (    v          )              2              ,where        d          G        (  u  ) is the degree of the vertex    u in    G. A cactus is a connected graph in which each block is either an edge or a cycle. Let        G    (  n  ,  k  ) be the set of cacti of order    n and with    k cycles. Obviously,        G    (  n  ,  0  ) is the set of all trees and        G    (  n  ,  1  ) is the set of all unicyclic graphs, then the cacti of order    n and with    k  (  k  ≥  2  ) cycles is a generalization of cycle number    k. In this paper, we establish a sharp upper bound for the Sombor index of a cactus in        G    (  n  ,  k  ) and characterize the corresponding extremal graphs. In addition, for the case when    n  ≥  6  k  −  3, we give a sharp lower bound for the Sombor index of a cactus in        G    (  n  ,  k  ) and characterize the corresponding extremal graphs as well. We also propose a conjecture about the minimum value of sombor index among        G    (  n  ,  k  ) when    n  ≥  3  k.}
}