On A Conjecture About Randi(?) Index

A conjecture about the minimum value of graphs with minimum degreeδwas proposed by Delorme, Favaron and Rautenbach [1]. Recently, Mustaphs Aouchiche and Pierre Hansen ([16]) showed that the conjecture does not hold in general by counterexamples and propose a modified conjecture. In this paper, we show the conjecture which proposed by Mustaphs Aouchiche and Pierre Hansen is true with minimum degree equal n-2 and n-3, respectively. And we give general counterexamples to disprove Conjecture 1 when n-2≥δ＞(n+1)/2 or n-2＞5＞(n+2)/2(n＞7). Finally, we modify the conjecture in [16] by our results. What's more, we show Delorme et al's conjecture is true for k-tree. At last, we give some properties of Randi(?) index for 2-tree of order n.