Properties of connected graphs having minimum degree distance

The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m ≥ n − 1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P 4 , contains a vertex z of degree n − 1 such that G − z has at most one connected component C such that | C | ≥ 2 and C has properties similar to those of G . y fixed k such that k = 0 , 1 or k ≥ 3 , if m = n + k and n ≥ k + 3 then the extremal graph is unique and it is isomorphic to K 1 + ( K 1 , k + 1 ∪ ( n − k − 3 ) K 1 ) .