Three steps away from Shapiro's problem: lower bounds for graphic sums with functions `max' or `min' in denominators

Abstract

Taking Shapiro's cyclic sums $\sum_{i=1}^n x_i/(x_{i+1}+x_{i+2})$ (assuming index addition mod $n$) as a starting point, we introduce a broader class of cyclic sums, called generalized Shapiro-Diananda sums, where the denominators are $p$-th order power means of the sets $\{x_{i+j_1},\dots,x_{i+j_k}\}$ with fixed distinct integers $j_1,\dots,j_k$ and $1\leq i\leq n$. Generalizing further, we replace the set of arguments of the power mean in the $i$-th denominator by an arbitrary nonempty subset of $\{1,\dots,n\}$ interpreted as the set of out-neighbors of the node number $i$ in a directed graph with $n$ nodes. We call such sums graphic power sums since their structure is controlled by directed graphs. The inquiry, as in the well-researched case of Shapiro's sums, concerns the greatest lower bound of the given ``sum'' as a function of positive variables $x_1,\dots,x_n$. We show that the cases of $p=+\infty$ (max-sums) and $p=-\infty$ (min-sums) are tractable. For the max-sum associated with a given graph the g.l.b. is always an integer; for a strongly connected graph it equals to graph's girth. For the similar min-sum, we could not relate the g.l.b. to a known combinatorial invariant; we only give some estimates and describe a method for finding the g.l.b., which has factorial complexity in $n$. A satisfactory analytical treatment is available for the secondary minimization -- when the g.l.b.'s of min-sums for individual graphs are mininized over the class of strongly connected graphs with $n$ nodes. The result (depending only on $n$) is found to be asymptotic to $e\ln n$.