F26A graph

The automorphism group of the F26A graph is a group of order 78.[3] 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, the F26A graph is the only cubic symmetric graph on 26 vertices.[2] It is also a Cayley graph for the dihedral group D₂₆, generated by a, ab, and ab⁴, where:[4]

${\displaystyle D_{26}=\langle a,b|a^{2}=b^{13}=1,aba=b^{-1}\rangle .}$

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).[5]

The characteristic polynomial of the F26A graph is equal to

${\displaystyle (x-3)(x+3)(x^{4}-5x^{2}+3)^{6}.\,}$

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.

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  The chromatic number of the F26A graph is 2.
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  The chromatic index of the F26A graph is 3.
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  Alternative drawing of the F26A graph.
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  F26A graph embedded in the torus.