continuing the work k. c. das, i. gutman, b. furtula, on second geometric-arithmetic indexof graphs, iran. j. math chem., 1(2) (2010) 17-28, in this paper we present lower and upperbounds on the third geometric-arithmetic index ga3 and characterize the extremal graphs.moreover, we give nordhaus-gaddum-type result for ga3.

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.

Let G be a simple graph with n vertices and m edges and Gc be its complement. Let δ(G) = δ and (G) = be the minimum degree and the maximum degree of vertices of G, respectively. In this paper, we present a sharp upper bound for the Laplacian spectral radius as follows: λ1(G) ( + δ − 1)+ √ ( + δ − 1)2 + 4(4m− 2δ(n− 1)) 2 . Equality holds if and only if G is a connected regular bipartite graph. A...