Decomposition Of Vertices Research Articles

A new two-variable generalization of the chromatic polynomial

We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph. This new polynomial satisfies both an edge decomposition formula and a vertex decomposition formula. We establish two general expressions for this new polynomial: one in terms of the broken circuit complex and one in terms of the lattice of forbidden colorings. We show that the new polynomial may be considered as a specialization of Stanley's chromatic symmetric function. We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees and graphs of restricted pathwidth.

Charge radius of the neutrino

Using the pinch technique we construct at one-loop order a neutrino charge radius, which is finite, depends neither on the gauge-fixing parameter nor on the gauge-fixing scheme employed, and is process independent. This definition stems solely from an effective proper photon-neutrino one-loop vertex, with no reference to box or self-energy contributions. The role of the $\mathrm{WW}$ box in this construction is critically examined. In particular it is shown that the exclusion of the effective $\mathrm{WW}$ box from the definition of the neutrino charge radius is not a matter of convention but is in fact dynamically realized when the target fermions are right-handedly polarized. In this way we obtain a unique decomposition of effective self-energies, vertices, and boxes, which separately respect electroweak gauge invariance. We elaborate on the tree-level origin of the mechanism which enforces at the one-loop level massive cancellations among the longitudinal momenta appearing in the Feynman diagrams, and in particular those associated with the non-Abelian character of the theory. Various issues related to the known connection between the pinch technique and the background field method are further clarified. Explicit closed expressions for the neutrino charge radius are reported.

Sequencing of instructional terms based on cluster analysis with weighted graph

We have proposed a structuring method of instructional terms and a “clustering method” with multiplicity on the basis of a weighted graph. In this paper we propose a consistent sequencing method of instructional terms based on them. Here we consider “level” and “relevancy (degree)” as the principle of the sequencing and give a set of rules for consistent sequencing with respect to them. They are represented as the rules on the adjacency matrix and a vertex decomposition method for it is discussed. We illustrate the method in some examples comparing the traditional method and make clear the properties of our method. Lastly we summarize the method by picturing a whole instructional strategy space. © 2000 Scripta Technica, Syst Comp Jpn, 31(4): 63–76, 2000

Embedded graph invariants in Chern-Simons theory

Chern-Simons gauge theory, since its inception as a topological quantum field theory, has proved to be a rich source of understanding for knot invariants. In this work the theory is used to explore the definition of the expectation value of a network of Wilson lines — an embedded graph invariant. Using a generalization of the variational method, lowest-order results for invariants for graphs of arbitrary valence and general vertex tangent space structure are derived. Gauge invariant operators are introduced. Higher order results are found. The method used here provides a Vassiliev-type definition of graph invariants which depend on both the embedding of the graph and the group structure of the gauge theory. It is found that one need not frame individual vertices. However, without a global projection of the graph there is an ambiguity in the relation of the decomposition of distinct vertices. It is suggested that framing may be seen as arising from this ambiguity — as a way of relating frames at distinct vertices.

Combining vertex decompositions with factoring in an all-terminal network reliability algorithm

The all-terminal network reliability problem is to determine the probability that all nodes can communicate in a network in which edges may fail independently of each other. One well known type of algorithm for treating this problem involves repeated deletion and contraction of edges in the network. These are commonly referred to as “factoring” algorithms and are especially effective on sparse graphs. On the other hand, a quite different approach known as the Buzacott algorithm is widely regarded as the most efficient algorithm for complete graphs. We demonstrate a manner in which higher degree vertex decomposition can be utilized within a factoring framework. This approach is compared to conventional factoring algorithms and the Buzacott algorithm from the standpoint of arithmetic operations performed and, in the case of the factoring algorithms, the search structures encountered during execution. Examples suggest that the higher degree vertex decompositions frequently reduce computational complexity to a greater extent than can be achieved by concentrating instead on an edge selection strategy to be used in factoring.

Analytic Decomposition of Vertices with Arbitrary Spin

Williams and Yao introduced a general method for decomposing vertex functions written as totally symmetric tensors. These particular vertex functions represent two on-shell particles of arbitrary spin with interger-spin difference coupling to bosons. In this paper we cover the case of half-integer-spin difference by letting the totally symmetric vertex function carry a free Dirac index. We write down all independent covariants consistent with Lorentz transformation properties of the vertex current and develop the constraint equations implicit in the Dirac algebra. This leads to an explicit decomposition into the proper number of independent vertex covariants. As Williams and Yao have shown, the decomposition is free from kinematic singularities. Although the method and results are general, we present explicitly the solutions that correspond to spin \textonehalf{}, $\frac{3}{2}$, and $\frac{5}{2}$.