13 problems

Consider the constructed families of undirected circulant graphs of dimension $5$, with degrees $10$ and $11$, whose orders are largest known and are given by quintic polynomials d…

Maximum-order conjecture. All these mixed graphs have the maximum possible order among mixed graphs with the same specified parameters.

Defect-2 nonexistence conjecture. There are no $(\Delta,D,-2)$-graphs with $\Delta\ge 4$ and $D\ge 4$.

Cyclic Cayley graph optimality conjecture. These graphs are optimal Abelian Cayley graphs of the specified type for every $k\geq 3$, not only for $3\leq k\leq 10$.

Three-generator optimality conjecture. The graphs given by that theorem are the largest undirected Cayley graphs of Abelian groups on three generators for each diameter $k$.

Three-generator Abelian Cayley graph conjecture. The constructed graph is as large as possible among undirected Cayley graphs of Abelian groups on three generators with diameter…

Let a $(d,k;+\epsilon)$-digraph be a $k$-geodetic digraph with minimum out-degree $d$ and order … where … Thus a $(d,k;+2)$-digraph has excess two. The conjecture on excess two dig…

Let a $k$-geodetic digraph be a directed graph in which, for every ordered pair of vertices, there is at most one walk of length at most $k$. If its minimum out-degree is $d$, its…

Let $AC(d,k)$ be the order of an extremal Abelian Cayley graph of degree $d$ and diameter $k$, and let $CC(d,k)$ be the order of an extremal circulant graph of degree $d$ and diame…

For an Abelian Cayley graph of degree $d$ and diameter $k$, let $f=\lfloor d/2\rfloor$. The conjectured coefficient relationships are known from comparisons of extremal and largest…

For an Abelian Cayley graph of degree $d$ and diameter $k$, let $f=\lfloor d/2\rfloor$. The order bounds and the known extremal and largest-known circulant families are compared th…

Abelian Cayley graph bound. For $k\ge2$ and $d\ge3$,

Let $d$ be an even degree, let $f=d/2$, and let $k$ be the diameter. Write the order of an extremal graph as a function of $d$ and $k$. Leading-term conjecture. For any even degree…