Doost Ali Mojdeh - Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
Iman Masoumi - Department of Mathematics, Tafresh University, Tafresh, Iran

For an integer k\geq۱, a k-distance enclaveless number (or k-distance B-differential) of a connected graph G=(V,E) is \Psi^k(G)=max\{|(V-X)\cap N_{k,G}(X)|:X\subseteq V\}. In this paper, we establish upper bounds onthe k-distance enclaveless number of a graph in terms of its diameter, radius and girth. Also, weprove that for connected graphs G and H with orders n and m respectively, \Psi^k(G\times H)\leq mn-n-m+\Psi^k(G)+\Psi^k(H)+۱, whereG\times H denotes the direct product of G and H.In the end of this paper, we show that the k-distance enclaveless number \Psi^k(T) of a tree T on n\geq k+۱vertices and with n_۱ leaves satisfies inequality \Psi^k(T)\leq\frac{k(۲n-۲+n_۱)}{۲k+۱}and we characterize the extremal trees.

k-distance enclaveless number, diameter, radius, girth, direct product