An Introduction to Closed/Open Neighborhood Sums: Minimax, Maximin, and Spread

Abstract

For a graph G of order |V(G)| = n and a real-valued mapping $${f:V(G)\rightarrow\mathbb{R}}$$ , if $${S\subset V(G)}$$ then $${f(S)=\sum_{w\in S} f(w)}$$ is called the weight of S under f. The closed (respectively, open) neighborhood sum of f is the maximum weight of a closed (respectively, open) neighborhood under f, that is, $${NS[f]={\rm max}\{f(N[v])|v \in V(G)\}}$$ and $${NS(f)={\rm max}\{f(N(v))|v \in V(G)\}}$$ . The closed (respectively, open) lower neighborhood sum of f is the minimum weight of a closed (respectively, open) neighborhood under f, that is, $${NS^{-}[f]={\rm min}\{f(N[v])|v\in V(G)\}}$$ and $${NS^{-}(f)={\rm min}\{f(N(v))|v\in V(G)\}}$$ . For $${W\subset \mathbb{R}}$$ , the closed and open neighborhood sum parameters are $${NS_W[G]={\rm min}\{NS[f]|f:V(G)\rightarrow W}$$ is a bijection} and $${NS_W(G)={\rm min}\{NS(f)|f:V(G)\rightarrow W}$$ is a bijection}. The lower neighbor sum parameters are $${NS^{-}_W[G]={\rm max}NS^{-}[f]|f:V(G)\rightarrow W}$$ is a bijection} and $${NS^{-}_W(G)={\rm max}NS^{-}(f)|f:V(G)\rightarrow W}$$ is a bijection}. For bijections $${f:V(G)\rightarrow \{1,2,\ldots,n\}}$$ we consider the parameters NS[G], NS(G), NS −[G] and NS −(G), as well as two parameters minimizing the maximum difference in neighborhood sums.