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A number of interesting graphs are associated with the work of van Cleemput and Zamfirescu (2018). Two 13- and 15-node graphs, denoted alpha and beta respectively, were used ...

Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line through ...

Cubic nonhamiltonian graphs are nonhamiltonian graphs that are also cubic. The numbers of connected cubic nonhamiltonian graphs on n=10, 12, ... nodes are 2, 5, 35, 219, ...

A figurate number of the form, CCub_n=n^3+(n-1)^3=(2n-1)(n^2-n+1). The first few are 1, 9, 35, 91, 189, 341, ... (OEIS A005898). The generating function for the centered cube ...

A pyramidal number of the form n(n+1)(5n-2)/6, The first few are 1, 8, 26, 60, 115, ... (OEIS A002413). The generating function for the heptagonal pyramidal numbers is ...

A figurate number which is equal to the cubic number n^3. The first few are 1, 8, 27, 64, ... (OEIS A000578).

A pyramidal number of the form n(n+1)(4n-1)/6, The first few are 1, 7, 22, 50, 95, ... (OEIS A002412). The generating function of the hexagonal pyramidal numbers is ...

A polyhedral graph corresponding to the skeleton of a Platonic solid. The five platonic graphs, the tetrahedral graph, cubical graph, octahedral graph, dodecahedral graph, ...

A quintic nonhamiltonian graph is a quintic graph that is nonhamiltonian. A number of such graphs are illustrated above. Owens (1980) showed that there exists a ...

A polyhedral graph is completely regular if the dual graph is also regular. There are only five types. Let rho be the number of graph edges at each node, rho^* the number of ...