Graph irreducible

Pictorially, graph g in Fig. 8 represents the low-density limit of Ac . Evaluation of Eq. (85) is much more difficult than for the chain contribution due to the presence of the additional association bond that closes the ring, rendering the graph irreducible irrespective of the superposition used to approximate (ij 4). To proceed, we consider the following simple superposition of g ... 

Eq. (55) is the sum of all simple irreducible diagrams that can be formed among the k nodes, every bond representing an ffj function. (Thus every node is multiply connected. If k = 2 then B is exceptional and corresponds to the graph of two nodes and an /l2> bond.) It should be noted that flf is zero when i and j coincide (cf. Eq. (44)). 

In Section IV, we develop the former results and we study the structure of the transport operator and of the generalized Boltzmann operator. We also analyse the irreducibility condition which appears in Prigogine s theory by using the graphs of equilibrium statistical mechanics. 

We study next the dynamical irreducibility condition which appeared in the definition of the transport operator. It eliminates from this quantity the reducible collision processes where the particles coming from infinity interact, recede to an infinite distance from one another, and then interact again. We define an extended transport operator from which the irreducibility condition is eliminated and which involves this time the reducible collisions. The relation between the transport operator and the extended transport operator is made explicit by means of a correspondence between the dynamical processes and the Mayer graphs for equilibrium. In this respect, we demonstrate, in these graphs, the importance of the role of the articulation points. 

The definition adopted here for this notion is different from that which was used in Section II (1,2-irreducible graphs). 

It is clear that two diagrams giving rise to the same graph may be reducible or not according to the order in which the interactions occur. For example, diagrams (a) and (b) of Fig. 7 are respectively reducible and irreducible even though one associates the same graph with them. 

Since we have not yet applied the condition of chronological separation, this expression still contains irreducible contributions —those in which, starting from the left, the first 8Lm) appears before the last 6L(n)—but they are easily eliminated so that the reducible contributions corresponding to our graph are given by the expression (see Eq. 58)... 

The irreducible contributions to generated by the two graphs with an articulation point are included in VF(423), which also contains all the contributions coming from the graph with three lines. The factor 1/— z in Eq. (74) comes from a propagator (66) with k = 0. 

We now have to apply the same methods to

articulation points (also those with more than four lines) and furnishes only irreducible terms (included in 1P(4234)). On the other hand, graph (b) contains reducible contributions. To obtain them, we again suppress in Eq. (70) the 8Lm corresponding to the... 

Loop expansion, vertex irreducible graphs and screened interaction... 

For example, the graphs 5.3a,b are vertex reducible. The graphs 5.3c. 5.1 are vertex irreducible. 

Fig. 5-4. An irreducible graph (first line) arid some reducible graphs (second line) yielding that graph after cutting off all reducible parts. The reducible parts are indicated by boxes...

Having reduced all quantities to their vertex irreducible parts we can introduce the screened interaction. In each order of the loop expansion we have to resum a series of diagrams. For instance, the series of one loop graphs... 

A category of reactions with a characteristic irreducible R-matrix is a set of basis reactions. The basis reactions correspond to the traditional classification of organic reactions . A basis reaction is best characterized in graph theoretical terms (ref. 13). The educts and the products of a basis reaction are expressed by a graph (see Fig. 7.2) whose nodes correspond to the reactive centers and whose lines indicate the bond orders of the covalent bonds that are directly affected by the reaction. The... 

Fig. 6. Feynman graphs responsible for the ln3(2Z)-2 contribution to the irreducible SESE a) graph in the low Z region. The graph a) yields the Karshenboim term, the graph b) corresponds to the additional Yerokhin term...

H-point functions are called irreducible if their diagrammatic expansions only consist of graphs which do not split into two pieces if one internal electron or photon line is cut. 

PjPj connecting P, to Pj. The resulting graph is said to be strongly connected if, for each pair (P, Pfi, there is a directed path P,Pa , Pk Pk, > Pr- Pj. A square matrix is irreducible if and only if its directed graph is strongly connected [LT, p. 529]. 

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