16 research outputs found
Independence numbers of some double vertex graphs and pair graphs
The combinatorial properties of double vertex graphs has been widely studied
since the 90's. However only very few results are know about the independence
number of such graphs. In this paper we obtain the independence numbers of the
double vertex graphs of fan graphs and wheel graphs. Also we obtain the
independence numbers of the pair graphs, that is a generalization of the double
vertex graphs, of some families of graphs.Comment: 17 pages. Minor changes in the proof
Independence and matching numbers of some token graphs
Let be a graph of order and let . The -token
graph of , is the graph whose vertices are the -subsets of
, where two vertices are adjacent in whenever their symmetric
difference is an edge of . We study the independence and matching numbers of
. We present a tight lower bound for the matching number of
for the case in which has either a perfect matching or an almost perfect
matching. Also, we estimate the independence number for bipartite -token
graphs, and determine the exact value for some graphs.Comment: 16 pages, 4 figures. Third version is a major revision. Some proofs
were corrected or simplified. New references adde
The packing number of the double vertex graph of the path graph
Neil Sloane showed that the problem of determine the maximum size of a binary
code of constant weight 2 that can correct a single adjacent transposition is
equivalent to finding the packing number of a certain graph. In this paper we
solve this open problem by finding the packing number of the double vertex
graph (2-token graph) of a path graph. This double vertex graph is isomorphic
to the Sloane's graph. Our solution implies a conjecture of Rob Pratt about the
ordinary generating function of sequence A085680.Comment: 21 pages, 7 figures. V2: 22 pages, more figures added. V3. minor
corrections based on referee's comments. One figure corrected. The title "On
an error correcting code problem" has been change
Some results on the laplacian spectra of Token graphs
This version of the contribution has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/978-3-030-83823-2_11. Use of this Accepted Version is subject to the publisher's Accepted Manuscript terms of use
http://www.spingernature.com/gp/open-research/policies/accepted-manuscript-terms.We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this work, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that 1=h=k=n2 , the Laplacian spectrum of Fh(G) is contained in the Laplacian spectrum of Fk(G) . Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement GÂŻÂŻÂŻÂŻ . This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs.The research of C. DalfÂŽo and M. A. Fiol has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The research of C. DalfÂŽo has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R. The research of C. Huemer was supported by PID2019-104129GBI00/ AEI/ 10.13039/501100011033 and Gen. Cat. DGR 2017SGR1336. F. J. Zaragoza MartŽınez acknowledges the support of the National Council of Science and Technology (Conacyt) and its National System of Researchers (SNI). This work has received funding from the European Unionâs Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 734922.Peer ReviewedPostprint (author's final draft
On the spectra and spectral radii of token graphs
Let be a graph on vertices. The -token graph (or symmetric -th
power) of , denoted by has as vertices the
-subsets of vertices from , and two vertices are adjacent when their
symmetric difference is a pair of adjacent vertices in . In particular,
is the Johnson graph , which is a distance-regular graph
used in coding theory. In this paper, we present some results concerning the
(adjacency and Laplacian) spectrum of in terms of the spectrum of .
For instance, when is walk-regular, an exact value for the spectral radius
(or maximum eigenvalue) of is obtained. When is
distance-regular, other eigenvalues of its -token graph are derived using
the theory of equitable partitions. A generalization of Aldous' spectral gap
conjecture (which is now a theorem) is proposed
Independence and matching number for some token graphs
Let G be a graph of order n and let k â {1, . . . , nâ1}. The k-token graph
Fk(G) of G is the graph whose vertices are the k-subsets of V (G), where
two vertices are adjacent in Fk(G) whenever their symmetric difference
is an edge of G. We study the independence and matching numbers of
Fk(G). We present a tight lower bound for the matching number of Fk(G)
for the case in which G has either a perfect matching or an almost perfect
matching. Also, we estimate the independence number for bipartite ktoken
graphs, and determine the exact value for some graphs
The automorphism groups of some token graphs
In this paper we obtain the automorphism groups of the token graphs of some
graphs. In particular we obtain the automorphism group of the -token graph
of the path graph , for . Also, we obtain the automorphism group
of the -token graph of the following graphs: cycle, star, fan and wheel
graphs.Comment: 18 pages, 8 figure
On the Laplacian spectra of token graphs
We study the Laplacian spectrum of token graphs, also called
symmetric powers of graphs. The k-token graph Fk(G) of
a graph G is the graph whose vertices are the k-subsets of
vertices from G, two of which being adjacent whenever their
symmetric difference is a pair of adjacent vertices in G. In this
paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show
that, for any integers h and k such that 1 †h †k †n
2 , the
Laplacian spectrum of Fh(G) is contained in the Laplacian
spectrum of Fk(G). We also show that the doubled odd
graphs and doubled Johnson graphs can be obtained as token
graphs of the complete graph Kn and the star Sn = K1,nâ1,
respectively. Besides, we obtain a relationship between the
spectra of the k-token graph of G and the k-token graph of
its complement G. This generalizes to tokens graphs a wellknown property stating that the Laplacian eigenvalues of G
are closely related to the Laplacian eigenvalues of G. Finally,
the doubled odd graphs and doubled Johnson graphs provide
two infinite families, together with some others, in which the
algebraic connectivities of the original graph and its token
graph coincide. Moreover, we conjecture that this is the case
for any graph G and its token graph.This research of C. DalfĂł and M.A. Fiol has been partially supported by AGAUR from the Catalan Government under project 017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. The research of C. DalfĂł has also been supported by MICINN from the Spanish Government under project MTM2017-83271-R. The research of C. Huemer was supported by MICINN from the Spanish Government under project PID2019-104129GB-I00/AEI/10.13039/501100011033 and AGAUR from the Catalan Government under project 017SGR1336. F.J. Zaragoza MartĂnez acknowledges the support of the National Council of Science and Technology (Conacyt) and its National System of Researchers (SNI). This research has also received funding from the European Unionâs Horizon 2020 research and innovation programme under the Marie SkĆodowska-Curie grant agreement No 73492