06000777

j

06000777 O'Neal, Allen Slater, Peter J. The minimax, maximin, and spread values for open neighborhood sums for 2-regular graphs. Congr. Numerantium 208, 19-32 (2011). 2011 Utilitas Mathematica Publishing Inc., Winnipeg EN graph labeling neighborhood sums 2-regular Summary: For a graph $G$ of order $|V(G)|= n$ and a real-valued mapping $f: V(G)\to \bbfR$, if $S\subset V(G)$ then $f(S)= \sum_{w\in S} f(w)$ is called the weight of $S$ under $f$. We define $NS[f]= \max\{f(N[v])|v\in V(G)\}$ and $NS(f)= \max\{f(N(v))|v\in V(G)\}$. Similarly, we define $NS^-[f]= \min\{f(N[v])|v\in V(G)\}$ and $NS^-(f)= \min\{f(N(v))|v\in V(G)\}$. For graph $G$ we define $NS[G] = \min\{NS[f]|f: V(G)\to [n]\text{ is a bijection}\}$, $NS(G) = \min\{NS(f)|f: V(G)\to [n]\text{ is a bijection}\}$, $NS^-[G] = \max\{NS^-[f]|f: V(G)\to [n]\text{ is a bijection}\}$ and $NS^-(G) = \max\{NS^-(f)|f: [n]\text{ is a bijection}\}$. The closed and open spread parameters of $G$ are $NS^{sp}[G] = \min\{NS[f]- NS^-[f]|f: V(g)\to [n]\text{ is a bijection}\}$ and $NS^{sp}(G) = \min\{NS(f)- NS^-(f)|f: V(G)\to [n]\text{ is a bijection}\}$. For cycles $C_k$ we present solutions for $NS(C_k)$, $NS^-(C_k)$, $NS^{sp}(C_n)$, and solutions for the union of equal sized cycles.