  Symmetric edge polytopes, also called adjacency polytopes, are lattice
polytopes determined by simple undirected graphs. We introduce the integer
array \(\mathrm{maxf}(n,m)\) giving the maximum number of facets of a symmetric
edge polytope for a connected graph having \(n\) vertices and \(m\) edges, and
the corresponding sequence \(\mathrm{minf}(n,m)\) of minimal values. We
establish formulas for the number of facets obtained in several classes of
sparse graphs and provide partial progress toward conjectures that identify
facet-maximizing graphs in these classes. These formulas are combinatorial in
nature and lead to independently interesting observations and conjectures
regarding integer sequences defined by sums of products of binomial
coefficients.
