In the
mathematical field of
graph theory, the F26A graph is a
symmetric bipartite
Bipartite may refer to:
* 2 (number)
* Bipartite (theology), a philosophical term describing the human duality of body and soul
* Bipartite graph, in mathematics, a graph in which the vertices are partitioned into two sets and every edge has an en ...
cubic graph with 26 vertices and 39 edges.
It has
chromatic number 2,
chromatic index 3,
diameter 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 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 H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle. The cycle ...
minus;7, 7sup>13.
Algebraic properties
The
automorphism group 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 (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 pairs of adjacent vertices and of , there is an automorphism
:f : V(G) \rightarrow V(G)
such that
:f(u_1) = u_2 and f(v_1) = v_2.
In ot ...
, the F26A graph is the only cubic symmetric graph on 26 vertices.
It is also a
Cayley graph for the
dihedral group ''D''
26, generated by ''a'', ''ab'', and ''ab''
4, where:
:
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 chara ...
of the F26A graph is equal to
:
Other properties
The F26A graph can be embedded as a
chiral regular map
in the torus, with 13 hexagonal faces. The
dual graph for this embedding is isomorphic to the
Paley graph of order 13.
Gallery
Image:F26A graph 2COL.svg , The chromatic number of the F26A graph is 2.
Image:F26A graph 3color edge.svg, The chromatic index 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.
References
{{reflist
Individual graphs
Regular graphs