Geodesic
Girth-regular graphs
Ars Mathematica Contemporanea, Tome 17 (2019) no. 2, pp. 349-368.

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We introduce a notion of a girth-regular graph as a k-regular graph for which there exists a non-descending sequence (a1, a2, …, ak) (called the signature) giving, for every vertex u of the graph, the number of girth cycles the edges with end-vertex u lie on. Girth-regularity generalises two very different aspects of symmetry in graph theory: that of vertex transitivity and that of distance-regularity. For general girth-regular graphs, we give some results on the extremal cases of signatures. We then focus on the cubic case and provide a characterisation of cubic girth-regular graphs of girth up to 5.
DOI : 10.26493/1855-3974.1684.b0d
Keywords: Graph, girth-regular, cubic, girth 
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Primož Potočnik; Janoš Vidali. Girth-regular graphs. Ars Mathematica Contemporanea, Tome 17 (2019) no. 2, pp. 349-368. doi : 10.26493/1855-3974.1684.b0d. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.1684.b0d/

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