We consider a simple graph (non-directed, without loops or multiple edges) $G=(V,E)$, where $V$ is the set of nodes and $E$ is the set of edges. The distance between two nodes is defined as the length of the shortest path between the nodes (the number of edges in this path). If there is no such path (for instance, where the nodes are in different connected components) then the distance is infinite. We denote the distance between the nodes $i\in V$ and $j \in V$ by $\mathrm{dist}(i,j)$. The decay centrality of rank $\delta \in (0,1)$ of node $i\in V$ is defined as $$ C_\delta(i) = \sum_{j \in V \backslash \{i\}} \delta^{\mathrm{dist}(i,j)}. $$
For node $i\in V$ we consider the following expression $$ f(G,i) = \sum_{j \in V \backslash \{i\}} \frac{\delta^{\mathrm{dist}(i,j)}}{\sum\limits_{k \in V \backslash \{j\}} \delta^{\mathrm{dist}(k,j)}}. $$ The value of the above expression at the central node (c) of a star network $S$ of $n$ nodes is $$ f(S,c) = \frac{n-1}{1+\delta(n-2)}. $$ Is the value $f(S,c)$ the upper bound of $f(G,i)$ for any simple graph?