$J^2$-Independence Parameters of Some Graphs

Let G be a graph. A subset I′ of a vertex-set V (G) of G is called a J2-independent in Gif for every pair of distinct vertices a, b ∈ I′, dG(a, b) ̸= 1, N2 G[a]\N2 G[b] ̸= ∅ and N2 G[b]\N2 G[a] ̸= ∅. The maximum cardinality among all J2-independent sets in G, denoted by αJ2 (G), is called the J2-independence number of G. Any J2-independent set I′satisfying |I′| = αJ2 (G) is called the maximum J2-independent set of G or an αJ2 -set of G. In this paper, we establish some boundsof this parameter on a generalized graph, join and corona of two graphs. We characterize J2-independent sets in some families of graphs, and we use these results to derive the exact values of parameters of these graphs. Moreover, we investigate the connections of this new parameter with other variants of independence parameters. In fact, we show that the J2-independence number of a graph is always less than or equal to the standard independence number.