1,001 research outputs found
Pushable chromatic number of graphs with degree constraints
Pushable homomorphisms and the pushable chromatic number of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph , we have , where denotes the oriented chromatic number of . This stands as first general bounds on . This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all , we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree lies between and which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when , we then prove that the maximum value of the pushable chromatic number is~ or~. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~ lies between~ and~. The former upper bound of~ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~
Homomorphisms of (n,m)-graphs with respect to generalised switch
An -graph has different types of arcs and different types of
edges. A homomorphism of an -graph to an -graph is a
vertex mapping that preserves adjacency type and directions. Notice that, in an
-graph a vertex can possibly have different types of neighbors.
In this article, we study homomorphisms of -graphs while an Abelian
group acts on the set of different types of neighbors of a vertex.Comment: 13 pages, conferenc
Spectral preorder and perturbations of discrete weighted graphs
In this article, we introduce a geometric and a spectral preorder relation on
the class of weighted graphs with a magnetic potential. The first preorder is
expressed through the existence of a graph homomorphism respecting the magnetic
potential and fulfilling certain inequalities for the weights. The second
preorder refers to the spectrum of the associated Laplacian of the magnetic
weighted graph. These relations give a quantitative control of the effect of
elementary and composite perturbations of the graph (deleting edges,
contracting vertices, etc.) on the spectrum of the corresponding Laplacians,
generalising interlacing of eigenvalues.
We give several applications of the preorders: we show how to classify graphs
according to these preorders and we prove the stability of certain eigenvalues
in graphs with a maximal d-clique. Moreover, we show the monotonicity of the
eigenvalues when passing to spanning subgraphs and the monotonicity of magnetic
Cheeger constants with respect to the geometric preorder. Finally, we prove a
refined procedure to detect spectral gaps in the spectrum of an infinite
covering graph.Comment: 26 pages; 8 figure
Volume of representation varieties
We introduce the notion of volume of the representation variety of a finitely
presented discrete group in a compact Lie group using the push-forward measure
associated to a map defined by a presentation of the discrete group. We show
that the volume thus defined is invariant under the Andrews-Curtis moves of the
generators and relators of the discrete group, and moreover, that it is
actually independent of the choice of presentation if the difference of the
number of generators and the number of relators remains the same. We then
calculate the volume of the representation variety of a surface group in an
arbitrary compact Lie group using the classical technique of Frobenius and
Schur on finite groups. Our formulas recover the results of Witten and Liu on
the symplectic volume and the Reidemeister torsion of the moduli space of flat
G-connections on a surface up to a constant factor when the Lie group G is
semisimple.Comment: 27 pages in AMS-LaTeX forma
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