On the difference of two generalized connectivities of a graph

The concept of k-connectivity $$\kappa '_{k}(G)$$ κ k ′ ( G ) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity $$\kappa _{k}(G)$$ κ k ( G ) , mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if $$\kappa '_k(G)\ne n-k+1$$ κ k ′ ( G ) ≠ n - k + 1 where $$k\ge 3$$ k ≥ 3 , then $$0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1$$ 0 ≤ κ k ′ ( G ) - κ k ( G ) ≤ n - k - 1 ; otherwise, $$-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k$$ - ⌊ k 2 ⌋ + 1 ≤ κ k ′ ( G ) - κ k ( G ) ≤ n - k . Moreover, all of these bounds are sharp. Some specific study is focused for the case $$k=3$$ k = 3 . As results, we characterize the graphs with $$\kappa '_3(G)=\kappa _3(G)=t$$ κ 3 ′ ( G ) = κ 3 ( G ) = t for $$t\in \{1, n-3, n-2\}$$ t ∈ { 1 , n - 3 , n - 2 } , and give a necessary condition for $$\kappa '_3(G)=\kappa _3(G)$$ κ 3 ′ ( G ) = κ 3 ( G ) by showing that for a connected graph G of order n and size m, if $$\kappa '_3(G)=\kappa _3(G)=t$$ κ 3 ′ ( G ) = κ 3 ( G ) = t where $$1\le t\le n-3$$ 1 ≤ t ≤ n - 3 , then $$m\le {n-2\atopwithdelims ()2}+2t$$ m ≤ n - 2 2 + 2 t . Moreover, the unique extremal graph is given for the equality to hold.