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An Eulerian graph is a graph containing an Eulerian cycle. The numbers of Eulerian graphs with , 2, ... nodes are 1, 1, 2, 3, 7, 15, 52, 236, ... (OEIS A133736), the first few of which are illustrated above.The corresponding numbers of connected Eulerian graphs are 1, 0, 1, 1, 4, 8, 37, 184, 1782, ... (OEIS A003049; Robinson 1969; Liskovec 1972; Harary and Palmer 1973, p. 117), the first few of which are illustrated above.Some care is needed in interpreting the term, however, since some authors define an Euler graph as a different object, namely a graph for which all vertices are of even degree (motivated by the following theorem). Euler showed (without proof) that a connected simple graph is Eulerian iff it has no graph vertices of odd degree (i.e., all vertices are of even degree). While the number of connected Euler graphs on nodes is equal to the number of connected Eulerian graphs on nodes, the counts are different for disconnected graphs since..

The term "Euler graph" is sometimes used to denote a graph for which all vertices are of even degree (e.g., Seshu and Reed 1961). Note that this definition is different from that of an Eulerian graph, though the two are sometimes used interchangeably and are the same for connected graphs.The numbers of Euler graphs with , 2, ... nodes are 1, 1, 2, 3, 7, 16, 54, 243, 243, 2038, ... (OEIS A002854; Robinson 1969; Mallows and Sloane 1975; Buekenhout 1995, p. 881; Colbourn and Dinitz 1996, p. 687), the first few of which are illustrated above. There is an explicit formula giving these numbers.There are more Euler graphs than Eulerian graphs since there exist disconnected graphs having multiple disjoint cycles with each node even but for which no single cycle passes through all edges. The numbers of Euler-but-not-Eulerian graphs on , 2, ... nodes are 0, 0, 0, 0, 0, 1, 2, 7, 20, 76, 334, 2498, ... (OEIS A189771), the first few of which are..

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)...

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.

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 .

"The" butterfly graph is a name sometimes given to the 5-vertex graph illustrated above. This graph is also known as the "bowtie graph" (West 2000, p. 12) and is the triangular snake graph . The butterfly graph is ungraceful (Horton 2003). It is implemented in the Wolfram Language as GraphData["ButterflyGraph"].A different type of butterfly graph is defined as follows. The -dimensional butterfly graph is a directed graph whose vertices are pairs , where is a binary string of length and is an integer in the range 0 to and with directed edges from vertex to iff is identical to in all bits with the possible exception of the th bit counted from the left.The -dimensional butterfly graph has vertices and edges, and can be generated in the Wolfram Language using ButterflyGraph[n, b] (with )...

The Shrikhande graph is a strongly regular graph on 16 nodes. It is cospectral with the rook graph , so neither of the two is determined by spectrum.The Shrikhande graph is the smallest distance-regular graph that is not distance-transitive (Brouwer et al. 1989, p. 136). It has intersection array .The Shrikhande graph is implemented in the WolframLanguage as GraphData["ShrikhandeGraph"].The Shrikhande graph has two generalized LCF notations of order 8, eleven of order 4, 53 of order 2, and 2900 of order 1. The graphs with LCF notations of orders four and eight are illustrated above.The Shrikhande graph appears on the cover of the book Combinatorial Matrix Theoryby Brualdi and Ryser (1991); illustrated above.The plots above show the adjacency, incidence, and graph distance matrices for the Shrikhande graph.It is an integral graph with graph spectrum .The bipartite double graph of the Shrikhandegraph is the Kummer graph.The..

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)

The cocktail party graph of order , also called the hyperoctahedral graph (Biggs 1993, p. 17) or Roberts graph, is the graph consisting of two rows of paired nodes in which all nodes but the paired ones are connected with a graph edge. It is the graph complement of the ladder rung graph , and the dual graph of the hypercube graph . It is the skeleton of the -cross polytope.This graph arises in the handshake problem. It is a complete n-partite graph that is denoted by Brouwer et al. (1989, pp. 222-223), and is distance-transitive, and hence also distance-regular.The cocktail party graph of order is isomorphic to the circulant graph . The -cocktail party graph is also the -Turán graph.Special cases are summarized in the following table.-cocktail party graph1empty graph 2square graph 3octahedral graph416-cell graphThe -cocktail party graph has independence polynomialwith corresponding recurrence equation..

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.

Grünbaum conjectured that for every , , there exists an -regular, -chromatic graph of girth at least . This result is trivial for or , but only a small number of other such graphs are known, including the 12-node Chvátal graph, 21-node Brinkmann graph, and 25-node Grünbaum graph. The Chvátal graph is illustrated above in a couple embeddings (e.g., Bondy; Knuth 2008, p. 39).It has 370 distinct (directed) Hamiltonian cycles, giving a unique generalized LCF notation of order 4 (illustrated above), two of order 6 (illustrated above), and 43 of order 1.The Chvátal graph is implemented in the WolframLanguage as GraphData["ChvatalGraph"].The Chvátal graph is a quartic graph on 12 nodes and 24 edges. It has chromatic number 4, and girth 4. The Chvátal graph has graph spectrum ...

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..

"The" octahedral graph is the 6-node 12-edge Platonic graph having the connectivity of the octahedron. It is isomorphic to the circulant graph , the cocktail party graph , the complete tripartite graph , and the 4-dipyramidal graph. Several embeddings of this graph are illustrated above.It is implemented in the Wolfram Languageas GraphData["OctahedralGraph"].The octahedral graph has 6 nodes, 12 edges, vertex connectivity 4, edge connectivity 4, graph diameter 2, graph radius 2, and girth 3. It is the unique 6-node quartic graph, and is also a quartic symmetric graph. It has chromatic polynomialand chromatic number 3. It is an integral graph with graph spectrum . Its automorphism group is of order .The octahedral graph is the line graph of the tetrahedralgraph.There are three minimal integral drawings of the octahedral graph, illustrated above, all with maximum edge length of 7 (Harborth and Möller 1994).The..

The rook graph (confusingly called the grid by Brouwer et al. 1989, p. 440) and also sometimes known as a lattice graph (e.g., Bouwer) is the graph Cartesian product of complete graphs, which is equivalent to the line graph of the complete bipartite graph . This is the definition adopted for example by Brualdi and Ryser (1991, p. 153), although restricted to the case . This definition corresponds to the connectivity graph of a rook chess piece (which can move any number of spaces in a straight line-either horizontally or vertically, but not diagonally) on an chessboard.The graph has vertices and edges. It is regular of degree , has diameter 3, girth 3 (for ), and chromatic number . It is also perfect (since it is the line graph of a bipartite graph) and vertex-transitive.The rook graph is also isomorphic to the Latin square graph. The vertices of such a graph are defined as the elements of a Latin square of order , with two vertices being adjacent..

In graph theory, a cycle graph , sometimes simply known as an -cycle (Pemmaraju and Skiena 2003, p. 248), is a graph on nodes containing a single cycle through all nodes. A different sort of cycle graph, here termed a group cycle graph, is a graph which shows cycles of a group as well as the connectivity between the group cycles. Cycle graphs can be generated in the Wolfram Language using CycleGraph[n]. Precomputed properties are available using GraphData["Cycle", n]. A graph may be tested to see if it is a cycle graph using PathGraphQ[g] && Not[AcyclicGraphQ[g]], where the second check is needed since the Wolfram Language believes cycle graphs are also path graphs (a convention which seems nonstandard at best).Special cases include (the triangle graph), (the square graph, also isomorphic to the grid graph ), (isomorphic to the bipartite Kneser graph ), and (isomorphic to the 2-Hadamard graph). The -cycle graph is isomorphic..

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