Geodesic
Multiplicities of sums in the explicit formulae for counting fixed length cycles in undirected graphs
Prikladnaâ diskretnaâ matematika, no. 4 (2011), pp. 42-55

Voir la notice de l'article provenant de la source Math-Net.Ru

An explicit formula for counting $k$-cycles in graphs is the combination of sums corresponding to the shapes of closed $k$-walks. It was shown that the maximum multiplicity of a sum in the formula is $[k/2]$ for, starting with $k=8$. In this work, we study the maximum sum multiplicity for some families of graphs: bipartite, triangle-free, planar, maximum vertex degree three, and their intersections. When $k$ is large, the biparticity and degree boundednesses are the only properties which decrease the maximum sum multiplicity by 1, providing $k\equiv2,3\pmod4$. Some combinations of properties in the case of $k\leq20$ yield the decrease by 1 or 2.
Keywords: counting cycles in graphs, shapes of closed walks, prism graphs. 
@article{PDM_2011_4_a5,
     author = {A. N. Voropaev},
     title = {Multiplicities of sums in the explicit formulae for counting fixed length cycles in undirected graphs},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {42--55},
     publisher = {mathdoc},
     number = {4},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2011_4_a5/}
}
TY  - JOUR
AU  - A. N. Voropaev
TI  - Multiplicities of sums in the explicit formulae for counting fixed length cycles in undirected graphs
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2011
SP  - 42
EP  - 55
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2011_4_a5/
LA  - ru
ID  - PDM_2011_4_a5
ER  - 
%0 Journal Article
%A A. N. Voropaev
%T Multiplicities of sums in the explicit formulae for counting fixed length cycles in undirected graphs
%J Prikladnaâ diskretnaâ matematika
%D 2011
%P 42-55
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2011_4_a5/
%G ru
%F PDM_2011_4_a5
A. N. Voropaev. Multiplicities of sums in the explicit formulae for counting fixed length cycles in undirected graphs. Prikladnaâ diskretnaâ matematika, no. 4 (2011), pp. 42-55. http://geodesic.mathdoc.fr/item/PDM_2011_4_a5/