Thomas Zaslavsky

Averaging Sets: A Generalization of Mean Values and Spherical Designs

: [0, 1 ] + R, one cannot of course expect there to be a single point x at which every one assumes its mean value, but there is an easy generalization in terms of weighted averages: there exist at most m points xi and positive weights oi summing to 1 so that \’ aifj(Xi) = I”fi dt for j = 1, 2 ,..., m.

Biased graphs. I. Bias, balance, and gains

A biased graph is a graph together with a class of cirles (simple closed paths), called balanced, such that no theta subgraph contains exactly two balanced circles. A gain graph is a graph in which each edge has a gain (a label from a group so that reversing the direction inverts the gain); a circle is balanced if its edge gain product is 1; this defines a biased graph. We initiate a series devoted to biased graphs and their matroids. Here we study properties of balance and also subgraphs and contractions of biased and gain graphs.

Biased graphs. II.: The three matroids

A biased graph Ω consists of a graph Γ and a class B of circles (simple, closed paths) in Γ, called balanced circles, such that no theta subgraph contains exactly two balanced circles. The bias matroid G(Ω) is a finitary matroid on the edge set E of Ω whose circuits are the balanced circles and the minimal connected edge sets of cyclomatic number two containing no balanced circle. We prove that these circuits define a matroid and we establish cryptomorphic definitions and other properties. Another finitary matroid on E, the lift matroid L(Ω), and its one-point extension the complete lift matroid L0(Ω), are obtained from the abstract matroid lift construction applied to the graphic matroid G(Γ) and the class B. The circuits of L(Ω) are the balanced circles and the minimal edge sets of cyclomatic number two (not necessarily connected) containing no balanced circle. We develop cryptomorphisms and other properties of L0(Ω) and L(Ω). There is no completely general construction rule, besides the bias and lift constructions, which assigns to each biased graph a matroid intermediate (in the sense of independent sets) between G and L and which respects subgraphs. G(Ω) has an infinitary analog for infinite graphs generalizing Klees infinitary bicircular matroid and the Bean-Higgs infinitary graphic matroid. Whether L(Ω) has an infinitary analog is unclear.

Signed graph coloring

Coloring a signed graph by signed colors, one has a chromatic polynomial with the same enumerative and algebraic properties as for ordinary graphs. New phenomena are the interpretability only of odd arguments and the existence of a second chromatic polynomial counting zero-free colorings. The generalization to voltage graphs is outlined.

Orientation of signed graphs

A graph with signed arcs is oriented by directing each end of each arc in accordance with a sign-compatibility rule. We prove that the regions of the hyperplane representation of a signed graph ∑ , as well as the vertices of the convex hull of all degree vectors of orientations of ∑ , are in natural one-to-one correspondence with the cyclic orientations of ∑ The proof uses the oriented matroid of a signed graph. For use elsewhere, we also develop the relationships between orientations and hyperplane representations of a signed graph and those of its double covering graph.

Characterizations of signed graphs

The possible classes of balanced circles of a signed graph are characterized in two ways. A signed graph is a graph with arcs signed f or -; a circle is balanced if the product of its arc signs is +. I give here two characterizations of the possible classes of balanced circles of a signed graph: an elementary one of the balanced portion of an arbitrary subclass of circles, and a stronger one of the entire balanced circle class. The latter characterizes signed graphs among biased graphs (explained in [9]). Terminology. A signed graph C consists of an ordinary graph r (finite or infinite) with node set N and arc set E , and a mapping u: E (+, -}, the sign labeling. Loops and multiple arcs are allowed (but we omit the half arcs and free loops needed in other parts of signed graph theory [S]). A path has a Value obtained by multiplying the signs of its constituent arcs; a circle whose value is + is called balanced. An arc set is called balanced when every circle in it is balanced. The class of circles of r is denotedqr); the class of circles balanced in C. is written B ( C ) . (Signed graphs and balance were first conceived by Harary [3].) See Figure 1 for illustrations of signed graphs. First Characterization. When is a class of circles equal t o g @ ) for some C? A generalization: given a certain class 9 of circles of r, when is a subclass @ equal to the balanced subclass o f 9 in some sign labeling of r? To solve the problem we look first at the binary vector space 9 of all subsets of E(T), whose addition is the symmetric difference A. I f 9 C9, we can speak of independent and spanning subsets of 53 (“spanning” means spanning 9). To say a subset 9 is additive in 53 means that whenever C, C1, ..., C, €9 and C = C1 A * A C,, then C €93 if and only if an even number of C , , . . . , C,. are not in 93. This is equivalent to saying that equals either 9 or the intersection of 9 with a hyperplane (codimension *Research assisted by support from the NSF and SGPNR Journal of Graph Theory, Vol. 5 (1981) 401-406 D 1981 by John Wiley & Sons, Inc. CCCC 0364-9024/81/040401-06

Strong Tutte functions of matroids and graphs

01 .OO 402 JOURNAL OF GRAPH THEORY

A combinatorial analysis of topological dissections

A strong Tutte function of matroids is a function of finite matroids which satisfies F(M 1 ○+M 2 ) = F(M 1 )F(M 2 ) and F(M)=a e F(M/e)+ b e F(M/e) for e not a loop or coloop of M, where a e , b e are scalar parameters depending only on e. We classify strong Tutte functions of all matroids into seven types, generalizing Brylawskis classification of Tutte-Grothendieck invariants. One type is, like Tutte-Grothendieck invariants, an evaluation of a rank polynomial; all types are given by a Tutte polynomial

The number of nowhere-zero flows on graphs and signed graphs

Abstract From a topological space remove certain subspaces (cuts), leaving connected components (regions). We develop an enumerative theory for the regions in terms of the cuts, with the aid of a theorem on the Mobius algebra of a subset of a distributive lattice. Armed with this theory we study dissections into cellular faces and dissections in the d -sphere. For example, we generalize known enumerations for arrangements of hyperplanes to convex sets and topological arrangements, enumerations for simple arrangements and the Dehn-Sommerville equations for simple polytopes to dissections with general intersection, and enumerations for arrangements of lines and curves and for plane convex sets to dissections by curves of the 2-sphere and planar domains.

Frame matroids and biased graphs

A nowhere-zero k-flow on a graph @C is a mapping from the edges of @C to the set {+/-1,+/-2,...,+/-(k-1)}@?Z such that, in any fixed orientation of @C, at each node the sum of the labels over the edges pointing towards the node equals the sum over the edges pointing away from the node. We show that the existence of an integral flow polynomial that counts nowhere-zero k-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the flow counting functions, and reciprocity laws that interpret the evaluations of the flow polynomials at negative integers in terms of the combinatorics of the graph.