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On balance index sets of one-point unions of graphs. (English) Zbl 1161.05063
Summary: Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A=\{0,1\}\). A labeling \(f:V(G)\to A\) induces an edge partial labeling \(f^*:E(G)\to A\) defined by \(f^*(xy)= f(x)\) if and only if \(f(x)=f(y)\) for each edge \(xy\in E(G)\). For each \(i\in A\), let \(v_f(i)= |\{v\in V(G): f(v)=i\}|\) and \(e_f(i)= |\{e\in E(G): f^*(e)=i\}\). The balance index set of \(G\), denoted \(\text{BI}(G)\), is defined as \(\{|e_f(0)- e_f(1)|: |v_f(0)-v_f(1)|\leq1\}\). In this paper, exact values of the balance index sets of five new families of one-point union of graphs are obtained, many of them, but not all, form arithmetic progressions.
05C78 Graph labelling (graceful graphs, bandwidth, etc.)