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 nn and size m≥n−1m≥n−1 are deduced. It is shown that any such graph GG has no induced subgraph isomorphic to P4P4, contains a vertex zz of degree n−1n−1 such that G−zG−z has at most one connected component CC such that |C|≥2|C|≥2 and CC has properties similar to those of GG.For any fixed kk such that k=0,1k=0,1 or k≥3k≥3, if m=n+km=n+k and n≥k+3n≥k+3 then the extremal graph is unique and it is isomorphic to K1+(K1,k+1∪(n−k−3)K1)K1+(K1,k+1∪(n−k−3)K1).