arXiv subjects

Joanna A. Ellis-Monaghan

Publications and source records attributed to Joanna A. Ellis-Monaghan.

24 records · Page 2Linked to original sources

arXiv2011

The Tutte-Potts connection in the presence of an external magnetic field

The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines. Unfortunately, it does not include the external magnetic fields that appear in most Potts model applications. Here we define the V-polynomial, which lifts the classical relationship between the Tutte polynomial and the zero field Potts model to encompass external magnetic fields. The V-polynomial generalizes Nobel and Welsh's W-polynomial, which extends the Tutte polynomial by incorporating vertex weights and adapting contraction to accommodate them. We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the V-polynomial, and hence a polynomial with deletion-contraction reduction and Fortuin-Kasteleyn type representation. This unifies an important segment of Potts model theory and brings previously successful combinatorial machinery, including complexity results, to bear on a wider range of statistical mechanics models.

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arXiv2010

Twisted duality for embedded graphs

We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|}, the ribbon group, on G. The action of the ribbon group on embedded graphs extends the concepts of duality, partial duality and Petrie duality. We show that this ribbon group action gives a complete characterization of duality in that if G is any cellularly embedded graph with medial graph G_m, then the orbit of G under the group action is precisely the set of all graphs with medial graphs isomorphic (as abstract graphs) to G_m. We provide characterizations of special sets of twisted duals, such as the partial duals, of embedded graphs in terms of medial graphs and we show how different kinds of graph isomorphism give rise to these various notions of duality. The ribbon group action then leads to a deeper understanding of the properties of, and relationships among, various graph polynomials via the generalized transition polynomial which interacts naturally with the ribbon group action.

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arXiv2009

A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan

In [A polynomial invariant of graphs on orientable surfaces, Proc. Lond. Math. Soc., III Ser. 83, No. 3, 513-531 (2001)] and [A polynomial of graphs on surfaces, Math. Ann. 323, 81-96 (2002)], Bollobas and Riordan generalized the classical Tutte polynomial to graphs cellularly embedded in surfaces, i.e. ribbon graphs, thus encoding topological information not captured by the classical Tutte polynomial. We provide a `recipe theorem' for their new topological Tutte polynomial, R(G). We then relate R(G) to the generalized transition polynomial Q(G) via a medial graph construction, thus extending the relation between the classical Tutte polynomial and the Martin, or circuit partition, polynomial to ribbon graphs. We use this relation to prove a duality property for R(G) that holds for both oriented and unoriented ribbon graphs. We conclude by placing the results of Chumutov and Pak [The Kauffman bracket and the Bollobas-Riordan polynomial of ribbon graphs, Moscow Mathematical Journal 7(3) (2007) 409-418] for virtual links in the context of the relation between R(G) and Q(R).

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arXiv2007

Unit Rectangle Visibility Graphs

Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more closely modeling the constrained dimensions of gates and other circuit components in computer chip applications. A graph $G$ is a unit rectangle visibility graph (URVG) if its vertices can be represented by closed unit squares in the plane with sides parallel to the axes and pairwise disjoint interiors, in such a way that two vertices are adjacent if and only if there is a non-degenerate horizontal or vertical band of visibility joining the two rectangles. Our results include necessary and sufficient conditions for $K_n$, $K_{m,n}$, and trees to be URVGs, as well as a number of general edge bounds.

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arXiv2006

Isotropic systems and the interlace polynomial

Through a series of papers in the 1980's, Bouchet introduced isotropic systems and the Tutte-Martin polynomial of an isotropic system. Then, Arratia, Bollobás, and Sorkin developed the interlace polynomial of a graph in [ABS00] in response to a DNA sequencing application. The interlace polynomial has generated considerable recent attention, with new results including realizing the original interlace polynomial by a closed form generating function expression instead of by the original recursive definition (see Aigner and van der Holst [AvdH04], and Arratia, Bollobás, and Sorkin [ABS04b]). Now, Bouchet [Bou05] recognizes the vertex-nullity interlace polynomial of a graph as the Tutte-Martin polynomial of an associated isotropic system. This suggests that the machinery of isotropic systems may be well-suited to investigating properties of the interlace polynomial. Thus, we present here an alternative proof for the closed form presentation of the vertex-nullity interlace polynomial using the machinery of isotropic systems. This approach both illustrates the intimate connection between the vertex-nullity interlace polynomial and the Tutte-Martin polynomial of an isotropic system and also provides a concrete example of manipulating isotropic systems. We also provide a brief survey of related work.

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arXiv2006

Distance Hereditary Graphs and the Interlace Polynomial

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors, evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, we prove that the one-variable vertex-nullity interlace polynomial is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan. We define the γinvariant as the coefficient of x^1 in the vertex-nullity interlace polynomial, analogously to the βinvariant, which is the coefficient of x^1 in the Tutte polynomial. We then turn to distance hereditary graphs, and show that graphs in this class have γinvariant of 2^{n+1} when n true twins are added in their construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γinvariant 2, just as the series-parallel graphs are exactly the class of graphs with βinvariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of any Euler circuit in the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distance hereditry graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

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