For a simple graph G and non-negative real number x, an L(x, 1)-labeling L of G assigns real-valued labels to V(G) such that the labels assigned to adjacent vertices differ by at least x and labels assigned to vertices at distance two differ by at least 1. The minimum span over all such labelings of G is denoted λ_(x,1)(G) and is called the λ_(x,1)-number of G. For integer r ≥ 2, the infinite r-path, denoted P_∞(r), is the graph with vertex set {v_i | i ∈ Z} andedge set {v_iv_j| |i - j| ≤ r - 1}. The r-path on n vertices is the graph inducedby v_1,v_2,...,v_n of the infinite r-path. This paper extends existing results on λ_(x,1) -numbers of various r-paths. For all x ≥ 0, we derive the λ_(x,1)-numbers of the infinite 3-path and infinite 4-path, and present partial results on the λ_(x,1)-number of all infinite r-paths for 0 ≤ x ≤ 1. For use in conjunction with certain theoretical results, a computer algorithm that determines the λ_(j,k)-number of graphs of small order for positive integers j,k is utilized to obtain the λ_(x,1)-numbers of 3-paths of all orders.
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