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Taking Shapiro's cyclic sums $\sum_{i=1}^n x_i/(x_{i+1}+x_{i+2})$ (assumingindex addition mod $n$) as a starting point, we introduce a broader class ofcyclic sums, called generalized Shapiro-Diananda sums, where the denominatorsare $p$-th order power means of the sets $\{x_{i+j_1},\dots,x_{i+j_k}\}$ withfixed 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 inthe $i$-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 directedgraph with $n$ nodes. We call such sums graphic power sums since theirstructure is controlled by directed graphs.  The inquiry, as in the well-researched case of Shapiro's sums, concerns thegreatest 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 aninteger; 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 knowncombinatorial invariant; we only give some estimates and describe a method forfinding the g.l.b., which has factorial complexity in $n$.  A satisfactory analytical treatment is available for the secondaryminimization -- when the g.l.b.'s of min-sums for individual graphs aremininized over the class of strongly connected graphs with $n$ nodes. Theresult (depending only on $n$) is found to be asymptotic to $e\ln n$.

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