Connected Graphs and Their Connectivities

Let m and n be positive integers with n − 1 ≤ m ≤ ( n 2 ) and CG(m,n) be the set of all non-isomorphic connected graphs of order n and size m. The vertex-connectivity and the edge-connectivity of a graph G are denoted by κ(G) and λ(G), respectively. We prove that if π ∈ {κ, λ}, then there exist positive integers a and b such that {π(G) : G ∈ CG(m,n)} = {x ∈ Z : a ≤ x ≤ b}. Thus {π(G) : G ∈ CG(m,n)} is uniquely determined by min(π;m,n) = min{π(G) : G ∈ CG(m,n)} and max(π;m,n) = max{π(G) : G ∈ CG(m,n)}. The values of min(π;m,n) and max(π;m,n) are obtained in all situations. AMS Subject Classification: 05C40