§ 2.1.4 Consider now E3 as a vector space, and choose any three of its vectors v0, v1, v2, imposing however that (v1 - v0) and (v2 - v0) be linearly independent:
k1(v1− v0) + k2(v2− v0) = 0 implies k1 = k2 = 0.
Defining a1 = k1, a2 = k2, a0 = −(k1+ k2), this is equivalent to saying that the two conditions
a0v0+ a1v1+ a2v2 = 0 a0 + a1+ a2 = 0
2.1. GRAPHS 53 imply a0 = a1 = a2 = 0. Such conditions ensure that no two vectors are colinear, that (v0, v1, v2) constitute a triad. Let us define a vector dependent on the triad (v0, v1, v2) by the two conditions
b =
2
X
i=0
bivi ;
2
X
i=0
bi = 1. (2.1)
The points determined by the barycentric coordinates bi describe a plane in E3 (Figure 2.3).
Figure 2.3: The barycentric coordinates.
§ 2.1.5 Suppose we consider only two of the vectors, say v0 and v1. They are of course linearly independent. We can in this case define, just as above, their dependent vectors by
e = b0v0+ b1v1 , b0+ b1 = 1.
Now, the coordinates b0 and b1 determine points on a straight line. In the same way, we can take only one of the vectors, say v0, and its dependent vector as v = v0 itself: this will determine a point.
Add now an extra condition on the dependent vectors: that each coor-dinate be strictly positive, bi > 0. Then, the vector b above will span the
interior of a triangle; the vector e will span an open edge; and v0 again will
“span” an isolated point, or a vertex. Notice that the coordinates related to the vector e give actually a homeomorphism between the interval (0, 1) and a line segment in E3, justifying the name open edge. If instead we allow bi ≥ 0, a segment homeomorphic to the closed interval [0, 1] results, a closed edge.
With these edges and vertices, graphs may now be defined as previously. An edge can be indicated by the pair (vi, vk) of its vertices, and a graph G by a set of vertices plus a set of pairs. An oriented graph is obtained when all the pairs are taken to be ordered pairs — which is a formal way of putting arrows on the edges. A path from vertex v1 to vertex vn+1 is a sequence of edges
¯
e1e¯2e¯3. . . ¯enwith ¯ei= (vi, vi+1) or ¯ei= (vi+1, vi). We have said that a graph is connected when, given two vertices, there exists at least one path connecting them. It is multiply-connected when there are at least two independent, non-intersecting paths connecting any two vertices. It is simply-connected when it is connected but not multiply-connected. In Physics, multiply-connected graphs are frequently called “irreducible” graphs.
§ 2.1.6 A path is not supposed to accord itself to the senses of the arrows.
A path is called simple if all its edges are distinct (one does not go twice through the same edge) and all its vertices are distinct except possibly v1 = vn+1. In this last case, it is a loop. An Euler path on G is a path with all edges distinct and going through all the vertices in G. The number nk of edges starting or ending at a vertex vk is its degree (“coordination number” would be more to the physicist’s taste; chemists would probably prefer “valence”).
Clearly the sum of all degrees on a graph is even, as PV
1 ni = 2E. The number of odd vertices (that is, those with odd degrees) is consequently even.
§ 2.1.7 The Bridges of K¨onigsberg Graph theory started up when Euler faced this problem. There were two islands in the river traversing Kant’s town, connected between each other and to the banks by bridges as in the scheme of Figure 2.4. People wanted to know whether it was possible to do a tour traversing all the bridges only once and finishing back at the starting point. Euler found that it was impossible: he reasoned that people should have a departure for each arrival at every point, so that all degrees should be even — which was not the case.
§ 2.1.8 Graph Theory has scored a beautiful victory for Mathematics with the recent developments on the celebrated (see Phys.3.2.5) four-color prob-lem. The old conjecture, recently “demonstrated”, was that four colors were sufficient for the coloring of a map. This is a problem of graph theory. As said above, graphs are also of large use in many branches of Physics. Through
2.1. GRAPHS 55
Figure 2.4: Scheme of downtown K¨onigsberg, and the corresponding graph.
Feynman’s diagram technique, they have a fundamental role as guidelines for perturbation calculations in field theory and in the many body problem.
In Statistical Mechanics, besides playing an analogous role in cluster expan-sions, graphs are basic personages in lattice models. In the Potts model, for example, where the underlying lattice can be any graph, they become entan-gled (sorry!) with knots (Phys.3.2.3). They also appear in the generalized use of the Cayley tree and the related Bethe lattice in approximations to more realistic lattice models (Phys.3.2.4).
§ 2.1.9 To the path ¯e1¯e2e¯3. . . ¯en we can associate a formal sum ε1e1+ ε2e2+ . . . + εnen ,
with εi = +1 if ¯ei = (vi, vi+1), and εi = −1 if ¯ei = (vi+1, vi). The sum is thus obtained by following along the path and taking the (+) sign when going in the sense of the arrows and the (-) sign when in the opposite sense. The sum is called the chain of the path and εi is the incidence number of ¯ei in the chain.
§ 2.1.10 A further formal step, rather gratuitous at first sight, is to gener-alize the εi’s to coefficients which are any integer numbers: a 1-chain on the graph G is a formal sum
P
imiei = m1e1+ m2e2+ . . . + mnen , with mj ∈ Z.
§ 2.1.11 We can define the sum of two 1-chains by P
imiei+P
im0jej =P
k(mk+ m0k)ek.
Calling “0” the 1-chain with zero coefficients (the zero 1-chain), the set of 1-chains of G constitutes an abelian group, the first order chain group on G, usually denoted C1(G). In a similar way, a 0-chain on G is a formal sum
r1v1+ r2v2+ . . . + rpvp ,
with rj ∈ Z. Like the 1-chains, the 0-chains on G form an abelian group, the zeroth chain group on G, denoted C0(G). Of course, C0(G) and C1(G) are groups because Z is itself a group: it was just to obtain groups that we have taken the formal step εi → mj above. Groups of chains will be seen to be of fundamental importance later on, because some of them will show up as topological invariants.
§ 2.1.12 Take the oriented edge ¯ej = (vj, uj). It is a 1-chain by itself. We define the (oriented) boundary of ¯ej as the 0-chain ∂ ¯ej = uj− vj. In the same way, the boundary of a general 1-chain is defined as
∂P
imiei =P
imi∂ei which is a 0-chain.
§ 2.1.13 The mapping
∂ : C1(G) −→ C0(G)
preserves the group operation and is called the boundary homomorphism. A 1-cycle on G is a loop, a closed 1-chain. It has no boundary and is formally defined as an element c ∈ C1(G) for which ∂c = 0 (the zero 0-chain). The set of 1-cycles on G form a subgroup, denoted Z1(G).
§ 2.1.14 Consider the examples of Figure 2.5: Take first the graph at the left in the figure: clearly,
∂(e1+ e2) = v3− v1,
∂(me1+ ne2) = nv3− mv1 + (m − n)v2. In the second graph,
∂(e1+ e2+ e3) = 2v3− 2v1.