Abstract
A simple graph with n vertices is called P i -connected if any two distinct vertices are connected by an elementary path of length i. In this paper, lower bounds of the number of edges in graphs that are both P 2- and P i-connected are obtained. Namely if i⩽ 1 2 (n+1) , then | E( G)|⩾((4 i−5)/(2 i−2))( n−1), and if i > 1 2 (n+ 1) , then | E( G)|⩾2( n−1) apart from one exeptional graph. Furthermore, extremal graphs are determined in the former.
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