20 research outputs found
Critical Groups of Graphs with Dihedral Actions
In this paper we consider the critical group of finite connected graphs which
admit harmonic actions by the dihedral group . In particular, we show that
if the orbits of the -action all have either or points then the
critical group of such a graph can be decomposed in terms of the critical
groups of the quotients of the graph by certain subgroups of the automorphism
group. This is analogous to a theorem of Kani and Rosen which decomposes the
Jacobians of algebraic curves with a -action
The Equivalence of Two Graph Polynomials and a Symmetric Function
The U-polynomial, the polychromate and the symmetric function generalization
of the Tutte polynomial due to Stanley are known to be equivalent in the sense
that the coefficients of any one of them can be obtained as a function of the
coefficients of any other. The definition of each of these functions suggests a
natural way in which to generalize them which also captures Tutte's universal
V-functions as a specialization. We show that the equivalence remains true for
the extended functions thus answering a question raised by Dominic Welsh.Comment: 17 page
On the evaluation of the Tutte polynomial at the points (1,-1) and (2,-1)
C. Merino proved recently the following identity between evaluations of the Tutte polynomial of complete graphs: t(; 1,−1) = t(;2,−1). In this work we extend this result by giving a large class of graphs with this property, that is, graphs G such that there exist two vertices u and v with t(G;1,−1) = t(G−{u,v};2,−1). The class is described in terms of forbidden induced subgraphs and it contains in particular threshold graphs.Postprint (published version
Some inequalities for the Tutte polynomial
We prove that the Tutte polynomial of a coloopless paving matroid is convex
along the portions of the line segments x+y=p lying in the positive quadrant.
Every coloopless paving matroids is in the class of matroids which contain two
disjoint bases or whose ground set is the union of two bases of M*. For this
latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a
>= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same
class of matroids. We also prove this conjecture for some families of graphs
and matroids.Comment: 17 page
On the structure of the h-vector of a paving matroid.
We give two proofs that the h-vector of any paving matroid is a pure 0-sequence, thus answering in the affirmative a conjecture made by Stanley, for this particular class of matroids. We also investigate the problem of obtaining good lower bounds for the number of bases of a paving matroid given its rank and number of elements.The first author was supported by Conacyt of México Proyect8397