(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let G be a graph. A sequence of distinct vertices Q=(a_1, a_2, …, a_n) of G is called an L-hop independent sequence if n=1 or if d_G(a_i,a_j) eq 2 for each i eq j, where i,j{1,2,…, n} and N_G[a_s] _t=1^s-1 N_G(a_t) eq for each s{2,…,n}. The L-hop independence number of G, denoted by α_Lh(G), is the maximum length among all L-hop independent sequences in G. This study explores and characterizes the L-hop independent sequences in some graphs, complementary prism of any graph, and in the join of two graphs. Some formulas and bounds of L-hop independence number with respect to the order of a graph and other parameters in graph theory are derived. Moreover, some relationships of L-hop independence with hop independence and legal hop independence are established.