Let k be a positive integer and G be a connected graph. The open k-neighborhoodset Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v} : dG(u, v) ≤ k}. A set S of vertices of G is a distance k- cost effective if for every vertex u in S, |Nk G(u) ∩ Sc| − |NkG(u) ∩ S| ≥ 0. The maximum cardinality of a distance k- cost effective set of G is called the upper distance k- cost effective number of G. In this paper, we characterized a distance k- cost effective set in the join of two graphs. As direct consequences, the bounds or the exact values of the upper distance k- cost effective numbers are determined.
Let G be a connected graph and k ≥ 1 be an integer. The open k-neighborhood Nk G(v) of v ∈ V (G) is the set Nk G(v) = {u ∈ V (G) \ {v}: dG(u, v) ≤ k}. A set S of vertices of G is called distance k-cost effective of G if for every vertex u in S, |Nk G(u) ∩ (V (G) \ S)| − |Nk G(u) ∩ S| ≥ 0. The maximum cardinality of a distance k-cost effective set of G is called the upper distance k-cost effective number of G. In this paper, we characterized the distance k-cost effective sets in the corona and lexicographic product of two graphs. Consequently, the bounds or the exact values of the upper distance k-cost effective numbers of these graphs are obtained.
In this note, a q-analogue of the non-central Stirling numbers of the first kind is defined by means of the limit of the exponential factorial. As a consequence, some combinatorial properties, such as triangular recurrence relation, vertical recurrence relation, horizontal recurrence relation and horizontal generating function are established. Moreover, the q-analogue is given interpretation in the counting of A-tableau, a special type of 0-1 tableau.
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