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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.
##### MSC:
 05C78 Graph labelling (graceful graphs, bandwidth, etc.)