In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, the F26A graph is a

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

bipartite

cubic graph In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bip ...

with 26 vertices and 39 edges. It has

chromatic number In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring i ...

chromatic index In graph theory, a proper edge coloring of a Graph (discrete mathematics), graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge colorin ...

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

5, radius 5 and

girth Girth may refer to: Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth ...

6. Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41–63, 2002. It is also a 3- vertex-connected and 3- edge-connected graph. The F26A graph is

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

and can be described by the

LCF notation In the mathematical field of graph theory, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by Harold Scott MacDonald Coxeter, H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain ...

minus;7, 7sup>13.

Algebraic properties

The

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

of the F26A graph is a group of order 78. It acts transitively on the vertices, on the edges, and on the arcs of the graph. Therefore, the F26A graph is a

symmetric graph In the mathematical field of graph theory, a graph is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices (u_1,v_1) and (u_2,v_2) of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 a ...

(though not distance transitive). It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the

Foster census In the mathematical field of graph theory, a graph is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices (u_1,v_1) and (u_2,v_2) of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 a ...

, the F26A graph is the only cubic symmetric graph on 26 vertices. It is also a

Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a Graph (discrete mathematics), graph that encodes the abstract structure of a group (mathematics), group. Its definition is sug ...

for the

dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

''D''₂₆, generated by ''a'', ''ab'', and ''ab''⁴, where: :

D_ = \langle a, b ,  a^2 = b^ = 1, aba = b^ \rangle .

The F26A graph is the smallest cubic graph where the automorphism group acts regularly on arcs (that is, on edges considered as having a direction).Yan-Quan Feng and Jin Ho Kwak, "One-regular cubic graphs of order a small number times a prime or a prime square," ''J. Aust. Math. Soc.'' 76 (2004), 345-35

The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

of the F26A graph is equal to :

(x-3)(x+3)(x^4-5x^2+3)^6. \,

Other properties

The F26A graph can be embedded as a

chiral Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is dist ...

regular map in the torus, with 13 hexagonal faces. The

dual graph In the mathematics, mathematical discipline of graph theory, the dual graph of a planar graph is a graph that has a vertex (graph theory), vertex for each face (graph theory), face of . The dual graph has an edge (graph theory), edge for each p ...

for this embedding is isomorphic to the

Paley graph In mathematics, Paley graphs are undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yiel ...

of order 13.

Gallery

Image:F26A graph 2COL.svg , The

of the F26A graph is 2. Image:F26A graph 3color edge.svg, The

of the F26A graph is 3. Image:F26A graph alt.svg, Alternative drawing of the F26A graph. File:F026 graph embedded in torus.svg, F26A graph embedded in the

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

References

{{reflist Individual graphs Regular graphs