in this paper, a new molecular-structure descriptor, the general sum–connectivity co–index  is considered, which generalizes the first zagreb co–index and the general sum–connectivity index of graph theory. we mainly explore the lower and upper bounds in termsof the order and size for this new invariant. additionally, the nordhaus–gaddum–type resultis also represented.

‎for a set of non-negative integers~$l$‎, ‎the $l$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $a_v subseteq {1,dots‎, ‎l}$ to vertices $v$‎, ‎such that every two vertices $u,v$ are adjacent if and only if $|a_u cap a_v|in l$‎. ‎the bipartite $l$-intersection number is defined similarly when the conditions are considered only for the ver...

In this paper we give sharp upper bounds on the Zagreb indices and characterize all trees achieving equality in these bounds. Also, we give lower bound on first Zagreb coindex of trees.

In this paper some upper and lower bounds for the greatest eigenvalues of the PI and vertex PI matrices of a graph G are obtained. Those graphs for which these bounds are best possible are characterized.

We give sharp upper bounds on the Zagreb indices and lower bounds on the Zagreb coindices of chemical trees and characterize the case of equality for each of these topological invariants.