On Friendly Index Sets of \((p, p+1)\)-Graphs

Let \( G \) be a simple graph. Any vertex labeling \( f: V(G) \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E(G) \to \mathbb{Z}_2 \) according to \( f^*(xy) = f(x) + f(y) \). For each \( i \in \mathbb{Z}_2 \), define \( v_f(i) = |\{v \in V(G) : f(v) = i\}| \), and \( e_f(i) = |\{e \in E(G) : f^*(e) = i\}| \). The friendly index set of the graph \( G \) is defined as \( \{|e_f(0) – e_f(1)| : |v_f(0) – v_f(1)| \leq 1\} \). We determine the friendly index sets of connected \( (p, p+1) \)-graphs with minimum degree \( 2 \). Many of them form arithmetic progressions. Those that are not miss only the second terms of the progressions.