Irreducible Graphs Research Articles

The one-loop five-graviton scattering amplitude and its low-energy limit

A covariant path-integral calculation of the even-spin structure contribution to the one-loop N-graviton scattering amplitude in the type-II superstring theory is presented. The apparent divergence of the N = 5 amplitude is resolved by separating it into twelve independent terms corresponding to different orders of inserting the graviton vertex operators. Each term is well defined in an appropriate kinematic region and can be analytically continued to physical regions where it develops branch cuts required by unitarity. The zero-slope limit of the N = 5 amplitude is performed, and the Feynman diagram content of the low-energy field theory is examined. Both one-particle irreducible (1PI) and one-particle reducible (1PR) graphs with massless internal states are generated in this limit. One set of 1PI graphs has the same divergent dependence on the cut-off as that found in the four-graviton case, and it is proved that such graphs exist for all N. The 1PR graphs are contributed by the poles in the world-sheet chiral Green functions.

Nonperturbative N-point functions from the gaussian effective action and renormalization of λφ 4

Recently, a covariant variational approach was developed to obtain a class of lower-bound approximations to Γ 1PI, the generating functional of one-particle irreducible graphs. The gaussian effective action (GEA) emerged in this context as a natural lower-bound to Γ 1PI. In this letter we renormalize the GEA, applying a scheme suggested by Consoli, based on the background field method. The renormalized GEA seems to indicate that, although SSB occurs, the shifted theory is free-as found by Consoli in a RG analysis of the effective potential. However, a functional expansion of the shifted GEA, involving the n-point functions, suggests that the particles of the theory are also massless.

A rigorous derivation of the gaussian representation for propagators in scalar field theories

Using the well-known α-representation for Feynman graph amplitudes in scalar ∅ 3 and ∅ 4 theories, we show that the α-parameter can everywhere on a given one-particle irreducible graph G with an infinite number of loops be taken equal to some constant α∼ 1 l where l is the number of propagators of G (once an integration over an over-all scale for all the α's is performed). This is done for any number of euclidean dimensions ( d<6 for ∅ 3 and d<4 for ∅ 4), except the one where such a theory is renormalizable.

Kuratowski-type theorems do not extend to pseudosurfaces

We show that, in general, graph embeddability in pseudosurfaces cannot be characterized by means of a finite set of forbidden (=irreducible) graphs.

Enumeration of labeled, connected, homeomorphically irreducible graphs

Enumeration of labeled, connected, homeomorphically irreducible graphs

Coincidence theorem for the direct correlation function of hard-particle fluids

The Mayerf-function for purely hard particles of arbitrary shape satisfiesf 2(1, 2)=−f(1, 2). This relation can be introduced into the graphical expansion of the direct correlation functionc(1, 2) to obtain a graphical expression for the case of exact coincidence, in position and orientation, of two identical hard cores. The resulting expression forc(1, 1)+1 contains only graphsG fromc(1), the sum of irreducible graphs with one labeled point. Relative to its coefficient inc(1),G occurs inc(1, 1) with an additional factorR c which is 1 for the leading graph in the expansion and of the form 2−2L(G) for all other graphs. HereL(G)=0, 1, 2,..., is a nonnegative integer. Topological analysis is used to derive an expression forL(G) in terms of the connectivity properties ofG.

Application of the linked cluster expansion to the n-component φ 4 theory

The linked cluster (high-temperature) expansion is worked out through 14th order for the O( n) symmetric φ 4 theory on a d-dimensional simple hypercubic lattice. The quantities expanded are the renormalized mass m R, the wave-function renormalization constant Z R and the renormalized coupling g R at zero momentum. The results obtained apply for all n and bare couplings, and for d = 2,3,4. Because a huge number of irreducible graphs (about 100 000) contribute to the order considered, a fully computerized approach was necessary, which we describe here in some detail. The series will be applied elsewhere to solve the 4-component model on a four-dimensional lattice.

Computing the genus of the 2-amalgamations of graphs

The above authors [2] and S. Stahl [3] have shown that if a graphG is the 2-amalgamation of subgraphsG1 andG2 (namely ifG=G1∪G2 andG1∩G2={x, y}, two distinct points) then the orientable genus ofG,γ(G), is given byγ(G)=γ(G1)+γ(G2)+e, wheree=0,1 or −1. In this paper we sharpen that result by giving a means by whiche may be computed exactly. This result is then used to give two irreducible graphs for each orientable surface.

Classes of interval graphs under expanding length restrictions

AbstractLet C(α) denote the finite interval graphs representable as intersection graphs of closed real intervals with lengths in [1, α]. The points of increase for C are the rational α ≥ 1. The set D(α) = [∩β&gt;αC(β)]\C(α) of graphs that appear as soon as we go past α is characterized up to isomorphism on the basis of finite sets E(α) of irreducible graphs for each rational α. With α = p/q and p and q relatively prime, ∣E(α)∣ is computed for all (p,q) with q ⩽ 2 and p = q + 1. When q = 1, E(p) contains only the bipartite star K1, p+2. A lowr bound on ∣E(α)∣ is given for all rational α.

On cubic graphs which are irreducible for nonorientable surfaces

Let ∑̃ n denote the nonorientable surface of nonorientable genus n. A graph G is irreducible for ∑̃ n if G does not embed in ∑̃ n but any proper subgraph does embed. Let I 3 ∑̃ n) denote the set of cubic irreducible graphs for ∑̃ n. This note proves Theorem. I 3 (∑̃ n) is finite for each n.

Regularized and renormalized Bethe-Salpeter equations: Some aspects of irreducibility and asymptotic completeness in renormalizable theories

Results on the links between 2-particle irreducibility and asymptotic completeness are presented in the framework of a renormalized Bethe-Salpeter formalism, introduced recently by J. Bros from an axiomatic viewpoint, for the most simple class of renormalizable theories. These results, which involve therenormalized 2-particle irreducible kernelG (i.e. from the perturbative viewpoint the sum of renormalized Feynman amplitudes of 2-particle irreducible graphs in the channel considered), complement the general quasi-equivalence previously established by Bros forregularized (non-renormalized) Bethe-Salpeter kernels. On the one hand, a formal derivation of (2-particle) asymptotic completeness from the irreducibility ofG is given. On the other hand, the links between regularized and renormalized kernels are investigated. This analysis provides in particular a converse derivation (up to some assumptions) of the 2-particle irreducibility ofG from asymptotic completeness. As a byproduct, it also provides a more explicit justification of previous heuristic derivations by K. Symanzik of integral equations betweenF and various differences of values ofG, and a simple alternative derivation of the recently proposed “renormalized” Bethe-Salpeter equation.

Lattice constant systems and some of their properties

Lattice constants are defined in a general way such that weak lattice constants, strong lattice constants, free multiplicities and coincidable occurrence factors are cases of the generalised lattice constants. A general method to express a lattice constant of one system as a linear combination of lattice constants of another system is described. Some properties of the conversion matrices are discussed. A systematic method to express the lattice constant of a reducible graph in terms of lattice constants of irreducible graphs is also studied.

What’s not inside a Cayley graph

There exist graphs of arbitrarily high girth and absolutely bounded degree which cannot be induced subgraphs of an irreducible Cayley graph.

The second Legendre transform of the P( φ) 2 generating functional

We prove the existence of the second Legendre transform of the generating functional of connected Green's functions for even weakly coupled P( φ) 2 models. The transformed functional is the generator of two particle irreducible graphs.

Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs

Recursive procedures are obtained for counting isomorphism classes of three-connected graphs and of two-connected graphs without vertices of degree 2. We apply an enumeration tool developed by R. W. Robinson to count non-isomorphic 2-connected graphs: he expressed the sums of cycle indices of automorphism groups of connected graphs in terms of those of their 2-connected components, and we do the same for 2-connected graphs and their 3-connected components.

Counting labelled three-connected and homeomorphically irreducible two-connected graphs

Labelled three-connected graphs and labelled two-connected graphs with no vertices of degree 2 are counted using methods similar to those used by Riddell to count labelled two-connected graphs.

A lower estimate for the achromatic number of irreducible graphs

The achromatic number of a graph is the largest number of independent sets its vertex set can be split into in such a way that the union of any two of these sets is not independent. A graph is irreducible if no two vertices have the same neighborhood. The achromatic number of an irreducible graph with n vertices is shown to be ≥( 1 2 −0(1))log n⧸ log logn, while an example of Erdös shows that it need not be log n/log 2+2 for any n. The proof uses an indiscernibility argument.

Λ-binding to nuclear matter in the fermi hypernetted chain approximation

The Fermi hypernetted chain technique (FHNC) has been extended to the problem of calculating the Λ-binding to nuclear matter. By using graphical techniques and considering the Λ-particle as an impurity, cluster expansions for ΛN and NN distribution functions have been obtained in terms of irreducible graphs. Topological analyses of the graphs have been made to obtain hypernetted chain equations for the two distribution functions for state-independent ΛN and NN correlation factors. Calculations are carried out for the Λ-binding energy including the rearrangement energy terms by employing central, state-independent and spin-averaged hard-core ΛN potentials. Results are in direct contradiction to the reaction matrix results as well as the empirical estimates.

Calculation of critical exponents toO(1n2)

Details of the calculation to $O(\frac{1}{{n}^{2}})$ of the polarization part and of the vertex function with one ${\ensuremath{\phi}}^{2}$ insertion are given for an $n$-component, three-dimensional system with $\mathrm{O}(n)$ symmetric, short-ranged interaction. The critical indices $\ensuremath{\lambda}$ and $\ensuremath{\mu}$ are deduced and are found to conform to other indices available to the same order. An analysis of how the requirements of exponentiation and universality are fulfilled shows that they are consequences of the fulfillment of a scaling law at the previous order. Cutoff-dependent contributions are seen to split off automatically. With possible higher-order calculations in mind considerable stress is put on the technical side: An apparently novel type of regularization technique allows one to pick up the relevant terms directly and is shown to be connected to a kind of analytic regularization. Graphs with diagonal insertions are calculated from generalized Feynman amplitudes associated with first-order diagrams. Those with vertex insertions, as well as irreducible graphs are handled by an integration technique taken over from conformal invariant field theory. The numerical performance of the $\frac{1}{n}$ expansion is discussed, and an attempt is made to obtain more meaningful numbers from the expansion through various Pad\'e- and Pad\'e---Borel-type resummations. The results are compared with information available from other sources.

The embeddings of a graph—A survey

AbstractTopological graph theory seeks to find answers to the question of how graphs map into surfaces. This paper surveys the information now available about the range of a graph, namely, the set of surfaces on which the graph can be “neatly” embedded. Several other closely related topics, such as irreducible graphs, coloring problems, and crossing numbers, are ignored. As is quite often the case with mathematical theories, this discipline developed in a rather haphazard manner. Many isolated results existed before the practitioners became aware of the fact that they were developing a theory. The turning point occurred in 1968, when Ringel and Youngs completed their proof of the Heawood conjecture. Their proof, in addition to settling an old unsolved problem, also reinforced the significance of the rotation systems. It is the author's belief that these rotation systems, together with the generalized embedding schemes can, and should, become the main tool in all investigations concerning the embeddings of a graph. This survey is written from that point of view. After defining the scope of the area surveyed, this paper proceeds to discuss the significance of the rotation systems and embedding schemes. Several theorems of a general nature are listed. Attention is then focused on the maximum and minimum genera of a graph. Discussion of the first of these is deferred to another survey article by R. Ringeisen to appear in a subsequent issue. The various methods developed by researchers in this area for determining the (minimum) genus are then described. This is followed by a listing of all the theoretical information that is available about the genus parameter. The paper includes two tables that exhibit most of the graphs with known genus.