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.