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### King graph

The king graph is a graph with vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a king.The number of edges in the king graph is , so for , 2, ..., the first few values are 0, 6, 20, 42, 72, 110, ... (OEIS A002943).The order graph has chromatic number for and for . For , 3, ..., the edge chromatic numbers are 3, 8, 8, 8, 8, ....King graphs are implemented in the Wolfram Language as GraphData["King", m, n].All king graphs are Hamiltonian and biconnected. The only regular king graph is the -king graph, which is isomorphic to the tetrahedral graph . The -king graphs are planar only for (with the case corresponding to path graphs) and , some embeddings of which are illustrated above.The -king graph is perfect iff (S. Wagon, pers. comm., Feb. 22, 2013).Closed formulas for the numbers of -cycles of with are given by(1)(2)(3)(4)where the formula for appears in Perepechko and Voropaev.The..

### Traceable graph

A traceable graph is a graph that possesses a Hamiltonian path. Hamiltonian graphs are therefore traceable, but the converse is not necessarily true. The numbers of traceable graphs on , 2, ... are 1, 1, 2, 5, 18, 91, 734, ... (OEIS A057864), where the singleton graph is conventionally considered traceable. The first few are illustrated above, with a Hamiltonian path indicated in orange for each.Every self-complementary graph is traceable(Clapham 1974; Camion 1975; Farrugia 1999, p. 52).The following table lists some named graphs that are traceable but not Hamiltonian.graph theta-0 graph7Petersen graph10Herschel graph11Blanuša snarks18flower snark20Coxeter graph28double star snark30Tutte's graph46Szekeres snark50McLaughlin graph276

### Queen graph

The queen graph is a graph with vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a queen. The -queen graphs have nice embeddings, illustrated above. In general, the default embedding with vertices corresponding to squares of the chessboard has degenerate superposed edges, the only nontrivial exception being the -queen graph.Queen graphs are implemented in the Wolfram Language as GraphData["Queen", m, n].The following table summarized some special cases of queen graphs.namecomplete graph tetrahedral graph The following table summarizes some named graph complements of queen graphs.-queen graph-knight graph-queen graph-queen graph-knight graphAll queen graphs are Hamiltonian and biconnected. The only planar and only regular queen graph is the -queen graph, which is isomorphic to the tetrahedral graph .The only perfect queen graphs are , , and .A closed formula..

### Pentatope

The pentatope is the simplest regular figure in four dimensions, representing the four-dimensional analog of the solid tetrahedron. It is also called the 5-cell, since it consists of five vertices, or pentachoron. The pentatope is the four-dimensional simplex, and can be viewed as a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The pentatope has Schläfli symbol .It is one of the six regular polychora.The skeleton of the pentatope is isomorphic to the complete graph , known as the pentatope graph.The pentatope is self-dual, has five three-dimensional facets (each the shape of a tetrahedron), 10 ridges (faces), 10 edges, and five vertices. In the above figure, the pentatope is shown projected onto one of the four mutually perpendicular three-spaces within the four-space obtained by dropping one of the four vertex components (R. Towle)...

### Harborth graph

The Harborth graph is the smallest known 4-regular matchstick graph. It is therefore both planar and unit-distance. It has 104 edges and 52 vertices. This graph was named after its discoverer H. Harborth, who first presented it to a general public in 1986 (Harborth 1994, Petersen 1996, Gerbracht 2006).The Harborth graph is implemented in the WolframLanguage as GraphData["HarborthGraph"].Analytic expressions for the vertices consisting of algebraic numbers of degree 22 (with large coefficients) were derived by Gerbracht (2006). As a consequence, Gerbracht (2006) also proved that the Harborth graph is rigid.

### Triangular snake graph

The triangular snake graph is the graph on vertices with odd defined by starting with the path graph and adding edges for , ..., . The first few are illustrated above, and special cases are summarized in the following table.1singleton graph 3triangle graph 5butterfly graphTriangular snakes are unit-distance and matchstick by construction, perfect. They are graceful when the number of triangles is congruent to 0 or 1 (mod 4) (Moulton 1989, Gallian 2018), which is equivalent to when .

### Torus grid graph

The torus grid graph is the graph formed from the graph Cartesian product of the cycle graphs and . is isomorphic to . can be formed starting with an grid graph and connecting corresponding left/right and top/bottom vertex pairs with edges. While such an embedding has overlapping edges in the plane, it can naturally be placed on the surface of a torus with no edge intersections or overlaps. Torus grid graphs are therefore toroidal graphs. The isomorphic torus grid graphs and are illustrated above.The torus grid graphs are quartic and Hamiltonianand have vertex count(1)Torus grid graphs are circulant graphs iff and are relatively prime, i.e., . In such cases, is isomorphic to . Special cases are summarized in the following table and illustrated above in attractive (but non-toroidal) embddings.graphcirculant graph generalized quadrangle quartic vertex-transitive graph Qt65tesseract graph Harary et al. (1973) conjectured that(2)for all..

### Hypercube graph

The -hypercube graph, also called the -cube graph and commonly denoted or , is the graph whose vertices are the symbols , ..., where or 1 and two vertices are adjacent iff the symbols differ in exactly one coordinate.The graph of the -hypercube is given by the graph Cartesian product of path graphs . The -hypercube graph is also isomorphic to the Hasse diagram for the Boolean algebra on elements.The above figures show orthographic projections of some small -hypercube graphs using the first two of each vertex's set of coordinates. Note that above is a projection of the usual cube looking along a space diagonal so that the top and bottom vertices coincide, and hence only seven of the cube's eight vertices are visible. In addition, three of the central edges connect to the upper vertex, while the other three connect to the lower vertex.Hypercube graphs may be computed in the Wolfram Language using the command HypercubeGraph[n], and precomputed properties..

### Desargues graph

The Desargues graph is the cubic symmetric graph on 20 vertices and 30 edges illustrated above in several embeddings. It is isomorphic to the generalized Petersen graph and to the bipartite Kneser graph . It is the incidence graph of the Desargues configuration. It can be represented in LCF notation as (Frucht 1976). It can also be constructed as the graph expansion of with steps 1 and 3, where is a path graph. It is distance-transitive and distance-regular graph and has intersection array .The Desargues graph is one of three cubic graphs on 20 nodes with smallest possible graph crossing number of 6 (the others being two unnamed graphs denoted CNG 6B and CNG 6C by Pegg and Exoo 2009), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).The Desargues is an integral graph with graph spectrum . It is cospectral with another nonisomorphic graph (Haemers and Spence 1995, van Dam and Haemers 2003).It is also a unit-distance..

### Disdyakis dodecahedron

The disdyakis dodecahedron is the dual polyhedron of the Archimedean great rhombicuboctahedron and Wenninger dual . It is also called the hexakis octahedron (Unkelbach 1940; Holden 1971, p. 55).If the original great rhombicuboctahedronhas unit side lengths, then the resulting dual has edge lengths(1)(2)(3)The inradius is(4)Scaling the disdyakis dodecahedron so that gives a solid with surface area and volume(5)(6)

### Flower snark

The flower snarks are a family of snarks discovered by Isaacs (1975) and denoted . is Tietze's graph, which is a "reducible snark" since it contains a cycle of length less than 5. is illustrated above in two embeddings, the second of which appears in Scheinerman and Ullman (2011, p. 96) as an example of a graph with edge chromatic number and fractional edge chromatic number (4 and 3, respectively) both integers but not equal. is maximally nonhamiltonian for odd (Clark and Entringer 1983).

### Meredith graph

The Meredith graph is a quartic graph on 70 nodes and 140 edges that is a counterexample to the conjecture that every 4-regular 4-connected graph is Hamiltonian.It is implemented in the Wolfram Languageas GraphData["MeredithGraph"].The Meredith graph has chromatic number 3 andedge chromatic number 5.The plots above show the adjacency, incidence,and distance matrices of the graph.

### Heawood graph

The Heawood graph is a cubic graph on 14 vertices and 21 edges which is the unique (3,6)-cage graph. It is also a Moore graph. The Heawood graph is also the generalized hexagon , and its line graph is the generalized hexagon . The Heawood graph is illustrated above in a number of embeddings.It has graph diameter 3, graph radius 3, and girth 6. It is cubic symmetric, nonplanar, Hamiltonian, and can be represented in LCF notation as .It has chromatic number 2 and chromaticpolynomialIts graph spectrum is .It is 4-transitive, but not 5-transitive (Harary 1994, p. 173).The Heawood graph is one of eight cubic graphs on 14 nodes with smallest possible graph crossing number of 3 (another being the generalized Petersen graph ), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).The Heawood graph corresponds to the seven-color torus map on 14 nodes illustrated above. The Heawood graph is the point/line incidence..

### Franklin graph

The Franklin graph is the 12-vertex cubic graph shown above whose embedding on the Klein bottle divides it into regions having a minimal coloring using six colors, thus providing the sole counterexample to the Heawood conjecture. The graph is implemented in the Wolfram Language as GraphData["FranklinGraph"].It is the 6-crossed prism graph.The minimal coloring of the Franklin graph is illustrated above.The Franklin graph is nonplanar but Hamiltonian. It has LCF notations and .The graph spectrum of the Franklin graph is .

### Tietze's graph

The cubic graph on 12 nodes and 18 edges illustrated above in a number of embeddings. It is a snark, albeit a trivial one by the usual definition of the snark.It is implemented in the Wolfram Languageas GraphData["TietzeGraph"].Tietze's graph is the unique almost Hamiltonian cubic graph on 12 vertices (Punnim et al. 2007). In fact, it is also maximally nonhamiltonian (Clark and Entringer 1983).Tietze's graph provides a 6-color coloring of the Möbiusstrip as illustrated above.The plots above show the adjacency, incidence, and graph distance matrices for Tietze's graph.

### Royle graphs

The Royle graphs are the two unique simple graphs on eight nodes whose sigma polynomials have nonreal roots (Read and Wilson 1998, p. 265). The sigma polynomials of these graphs are given by(1)(2)respectively, each of which has two nonreal roots (and where the members of each pairs are complex conjugates of each other).The Royle graphs are implemented in the Wolfram Language as GraphData["RoyleGraph1"] and GraphData["RoyleGraph2"].The numbers of simple graphs having this property on , 2, ... vertices are 0, 0, 0, 0, 0, 0, 0, 2, 42, ..., with the 42 such graphs on 9 vertices illustrated above.

### Gr&uuml;nbaum graphs

Grünbaum conjectured that for every , , there exists an -regular, -chromatic graph of girth at least . This result is trivial for and , but only a small number of other such graphs are known, including the Grünbaum graph, illustrated above, Brinkmann graph, and Chvátal graph.The Grünbaum graph can be constructed from the dodecahedral graph by adding an additional ring of five vertices around the perimeter and cyclically connecting each new vertex to three others as shown above (left figure). A more symmetrical embedding is shown in the center figure above, and an LCF notation-based embedding is shown in the right figure. This graph is implemented in the Wolfram Language as GraphData["GruenbaumGraph25"].The Grünbaum graph has 25 vertices and 50 edges. It is a quartic graph with chromatic number 4, and therefore has . It has girth .It has diameter 4, graph radius 3, edge connectivity 4, and vertex connectivity..

### Caveman graph

The (connected) caveman graph is a graph arising in social network theory formed by modifying a set of isolated -cliques (or "caves") by removing one edge from each clique and using it to connect to a neighboring clique along a central cycle such that all cliques form a single unbroken loop (Watts 1999). A number of cavemen graphs formed in this manner from are illustrated above.Caveman graphs are perfect.Caveman graphs will are implemented in the Wolfram Language as GraphData["Caveman", n, k].

The -tadpole graph, also called a dragon graph (Truszczyński 1984) or kite graph (Kim and Park 2006), is the graph obtained by joining a cycle graph to a path graph with a bridge.The -tadpole graph is sometimes known as the -pan graph. The particular cases of the - and -tadpole graphs are also known as the paw graph and banner graph, respectively (ISGCI).Precomputed properties of tadpole graphs are available in the Wolfram Language as GraphData["Tadpole", m, n].Koh et al. (1980) showed that -tadpole graphs are graceful for , 1, or 3 (mod 4) and conjectured that all tadpole graphs are graceful (Gallian 2018). Guo (1994) apparently completed the proof by filling in the missing case in the process of showing that tadpoles are graceful when or 2 (mod 4) (Gallian 2018).

### Tur&aacute;n graph

A Turán graph, sometimes called a maximally saturated graph (Zykov 1952, Chao and Novacky 1982), with positive integer parameters and is a type of extremal graph on vertices originally considered by Turán (1941). There are unfortunately two different conventions for the index .In the more standard terminology (and that adopted here), the -Turán graph, sometimes also called a K-graph and variously denoted , (Gross and Yellen 2006, p. 476), (Chao and Novacky 1982), or (Pach and Agarwal 1995, p. 120), is the extremal graph on graph vertices that contains no -clique for (Chao and Novacky 1982; Diestel 1997, p. 149; Bollobás 1998, p. 108). In other words, the Turán graph has the maximum possible number of graph edges of any -vertex graph not containing a complete graph . The Turán graph is also the complete -partite graph on vertices whose partite sets are as nearly equal in..

### Circulant graph

A circulant graph is a graph of graph vertices in which the th graph vertex is adjacent to the th and th graph vertices for each in a list . The circulant graph gives the complete graph and the graph gives the cyclic graph .The circulant graph on vertices on an offset list is implemented in the Wolfram Language as CirculantGraph[n, l]. Precomputed properties are available using GraphData["Circulant", n, l].With the exception of the degenerate case of the path graph , connected circulant graphs are biconnected, bridgeless, cyclic, Hamiltonian, LCF, regular, traceable, and vertex-transitive.A graph is a circulant iff the automorphism group of contains at least one permutation consisting of a minimal cycle of length .The numbers of circulant graphs on , 2, ... nodes (counting empty graphs as circulant graphs) are 1, 2, 2, 4, 3, 8, 4, 12, ... (OEIS A049287), the first few of which are illustrated above. Note that these numbers cannot be counted..

### Sylvester graph

"The" Sylvester graph is a quintic graph on 36 nodes and 90 edges that is the unique distance-regular graph with intersection array (Brouwer et al. 1989, §13.1.2; Brouwer and Haemers 1993). It is a subgraph of the Hoffman-Singleton graph obtainable by choosing any edge, then deleting the 14 vertices within distance 2 of that edge.It has graph diameter 3, girth 5, graph radius 3, is Hamiltonian, and nonplanar. It has chromatic number 4, edge connectivity 5, vertex connectivity 5, and edge chromatic number 5.It is an integral graph and has graph spectrum (Brouwer and Haemers 1993).The Sylvester graph of a configuration is the set of ordinarypoints and ordinary lines.

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