Graph Theory.
2. Vertex Descriptors and Graph Coloring
Lorentz JÄNTSCHI
Technical University of Cluj-Napoca
http://lori.academicdirect.ro
Abstract
This original work presents the construction of a set of ten sequence matrices and their applications for ordering vertices in graphs. For every sequence matrix three ordering criteria are applied: lexicographic ordering, based on strings of numbers, corresponding to every vertex, extracted as rows from sequence matrices; ordering by the sum of path lengths from a given vertex; and ordering by the sum of paths, starting from a given vertex.
We also examine a graph that has different orderings for the above criteria. We then proceed to demonstrate that every criterion induced its own partition of graph vertex.
We propose the following theoretical result: both LAVS and LVDS criteria generate identical partitioning of vertices in any graph.
Finally, a coloring of graph vertices according to introduced ordering criteria was proposed.
Keywords
Graph theory, Vertex descriptors, Matrix based descriptors, Invariants, Graph coloring, Graph partitioning.
1. Introduction
The problem of vertex partitioning in graphs can be solved in different ways [1],[2]. The most frequently used discriminating criteria are well-known square matrices that collect contributions of pairs of vertices from the graph. The adjacency matrix is the simplest matrix of this type. Some authors use characteristic polynomial and its eigenvalue to characterize a given graph [3]. The sequence of eigenvalues forms the graph spectrum [4]. A basic result in spectral geometry is the theorem of Cheng [5].
Let us consider a sequence matrix. It is of dimension n×(n-1), where n is the number of vertices. Such a matrix can be used to compare different graphs (interstructural similarity), when analyzing the columns of sequence matrices, the sums by columns or global sums (generic indices) [6]. It is also possible to compare the vertices of a graph (intrastructural similarity), when analyzing the rows of sequence matrices and the sums by rows [7],[8]. With this observation, a sequence matrix is an elegant method to investigate graphs [9]. Note that the ordering of graph vertices by using sequence matrices is of a partially ordered type [10],[11].
Prior, Dobrynin reported the degeneracy of some sequence matrices [12],[13]. Decomposition of graphs into congruent factors has interesting chemical implications [14],[15]. Dobrynin [16] and Diudea [17],[18] reported distance degree sequences (DPS in our notation) and their chemical applications. The path sequence matrix (APS in our notation) is calculated in order to study molecular similarity [19].
2. Theoretical Considerations
Some definitions are required. Let G = (V, E) be an un-oriented graph. We are note the set of all path in G with P(G). The distance from i to j in G is d(G)i,j and detour is δ(G)i,j. For p Î P(G), l(p) denotes the length of path p.
t = (vi)1≤i≤n is a terminal path if:
(1) t Î P(G) and
(2) "v Î V s. th. (vn, v) Î E then t È (vn, v) Ï P(G).
The set of terminal paths starting with i in G is T(G)i = {t = (vi)1≤i≤n s. th. v1 = i and t terminal path}. The set of terminal paths in G is T(G) = (T(G)v)vÎV.
The set of distance
paths from i to j in G is DP(G)i,j = {p=(xi)1£i£n s. th. pÎP(G), x1=i, xn=j & l(p)=d(G)i,j}.
The set of distance path in G is DP(G) = {p = (xi)1£i£n s. th. p Î P(G) and l(p) = 
}.
The set of detour
paths from i to j in G is ΔP(G)i,j = {p=(xi)1£i£n s. th. pÎP(G), x1=i, xn=j & l(p)=δ(G)i,j}.
The set of detour path in G is ΔP(G) = {p = (xi)1£i£n s. th. p Î P(G) and l(p) = 
}.
The set of distance terminal paths from i to j in G is DT(G)i,j = {t=(xi)1£i£n s. th. tÎT(G), x1=i, xn=j & l(t)=d(G)i,j}.
The set of distance
terminal path in G
is DT(G) = {t = (xi)1£i£n s. th. t Î T(G) and l(t) = }.
The set of detour terminal paths from i to j in G is ΔT(G)i,j = {t=(xi)1£i£n s. th. tÎT(G), x1=i, xn=j & l(t)= δ(G)i,j}.
The set of detour
terminal path in G
is ΔT(G) = {t = (xi)1£i£n s. th. t Î T(G) and l(t) = }.
Note that if G = (V, E) is a connected graph, then " v Î V, " e Î E $ t Î T(G)v s. th. e Í t. Thus, any connected graph can be reconstructed from the terminal path list starting from an arbitrary vertex. Also, " i, j Î V, T(G)j can be constructed from T(G)i. Consequently, the sets of terminal paths T(G)v, v Î V are equivalent by construction.
In order to generate sequence matrices, we compute terminal paths. Let G = (V, E), V = {1, …, 8}, E = {(1, 2), (1, 3), (2, 4), (3, 5), (3, 6), (4, 5), (4, 7), (6, 7), (7,8)} shown in figure 1:
Fig. 1. Graph G (see above)
We have devised a 16-bit windows original computer program to list all terminal paths. For the graph in figure 1, we have obtained the following set of terminal paths for vertex 1:
T(G)1
1 2 4 5 3 6 7 8
1 2 4 7 8
1 2 4 7 6 3 5
1 3 6 7 8
1 3 6 7 4 2
1 3 6 7 4 5
1 3 5 4 7 6
1 3 5 4 7 8
1 3 5 4 2
Fig. 2. Terminal paths of graph G from fig. 1 starting from vertex 1
3. The Construction of Sequence Matrices of Numbering
We extend the concept used for the distance degree sequence matrix16 and layer path (path degree sequence)19 matrix to construct a set of 10 sequence matrices. A sequence matrix is a matrix that collects vertex contribution in each row and elongation of paths in each column. The elements of the matrix cumulate a value of vertex property for a given elongation. Sequence matrices are of dimension n×(n-1) and are given by following definitions:
SMN (sequence matrix of numbering) of paths (P) which starts with i and has length j is APS(G)i,j = |{p Î P(G), s. th. p1 = i, l(p) = j}|, where p1 is the first vertex from path p and operator |∙| is the cardinal operator.
SMN of distance P that starts with i and has length j is DPS(G)i,j = |{p Î APS(G)i,j, s. th. p Î DP(G)}|.
SMN of detour P that starts with i and has length j is ΔPS(G)i,j = |{p Î APS(G)i,j, s. th. p Î ΔP(G)}|.
SMN of terminal paths (T) which starts with i and has length j is ATS(G)i,j = |{t Î T(G), s. th. t1 = i, l(t) = j}}|.
SMN of distance T that starts with i and has length j is DTS(G)i,j = |{p Î ATS(G)i,j, s. th. p Î DT(G)}|.
SMN of detour T that starts with i and has length j is ΔTS(G)i,j = |{p Î ATS(G)i,j, s. th. p Î ΔT(G)}|.
SMN of the vertex situated at the end of paths that starts with i and has j edges is AVS(G)i,j = |{vÎV, s. th. $ p Î P(G), p1 = i, pj+1 = v, l(p) = j}|.
SMN of the vertex situated at the end of distance paths that starts with i and has j edges is DVS(G)i,j = |{vÎV, s. th. $ pÎDP(G), p1 = i, pj+1 = v, l(p)=j}|.
SMN of the vertex situated at the end of detour paths that starts with i and has j edges is ΔVS(G)i,j = |{vÎV, s. th. $ pÎΔP(G), p1 = i, pj+1 = v, l(p)=j}|.
SMN of the cycles that contain vertex i and has j edges is ACS(G)i,j = |{p Î P(G), s. th. l(p) = j-1 and (pj,p1) Î E}|.
If an lexicographically ordering are applied to the sequence matrices of numbering (SMN), through permutation of one line with another, and the new matrix obtained can be used to make interstructural comparisons (with any other graph) because a lexicographically ordered matrix is independent with respect to labeling.
When two vertices are compared, lexicographic ordering is also useful from the point of view of the selected criterion.
The results of lexicographic ordering can be used to index the graph, by permuting the old indexing.
The computer program used in lexicographic ordering (quicksort method), gives permutation of partial ordering (σ in what follows).
Ten sequence matrices, ten lexicographically associated matrices and ten permutations σ are computed.
Two vertex indices in every case (from ten cases) are calculated. The first is sum of elements from the row (the number of total paths), the second is the weighted sum of elements from the row (the length of total paths).
For the graph G from figure 1 we obtain a set of ten tables with three different ordering criteria (table 1-10).
Table 1. APS and LAPS numbered sequences
|
APS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LAPS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
2 |
3 |
4 |
7 |
6 |
2 |
1 |
25 |
97 |
|
1=σ(8) |
1 |
2 |
3 |
4 |
5 |
5 |
2 |
22 |
99 |
|
2 |
2 |
3 |
5 |
5 |
6 |
4 |
0 |
25 |
97 |
|
2=σ(1) |
2 |
3 |
4 |
7 |
6 |
2 |
1 |
25 |
97 |
|
3 |
3 |
3 |
5 |
7 |
3 |
0 |
0 |
21 |
67 |
|
3=σ(2) |
2 |
3 |
5 |
5 |
6 |
4 |
0 |
25 |
97 |
|
4 |
3 |
4 |
4 |
6 |
3 |
1 |
0 |
21 |
68 |
|
4=σ(6) |
2 |
4 |
4 |
5 |
6 |
3 |
0 |
24 |
90 |
|
5 |
2 |
4 |
5 |
5 |
4 |
5 |
1 |
26 |
102 |
|
5=σ(5) |
2 |
4 |
5 |
5 |
4 |
5 |
1 |
26 |
102 |
|
6 |
2 |
4 |
4 |
5 |
6 |
3 |
0 |
24 |
90 |
|
6=σ(7) |
3 |
3 |
4 |
5 |
5 |
2 |
0 |
22 |
78 |
|
7 |
3 |
3 |
4 |
5 |
5 |
2 |
0 |
22 |
78 |
|
7=σ(3) |
3 |
3 |
5 |
7 |
3 |
0 |
0 |
21 |
67 |
|
8 |
1 |
2 |
3 |
4 |
5 |
5 |
2 |
22 |
99 |
|
8=σ(4) |
3 |
4 |
4 |
6 |
3 |
1 |
0 |
21 |
68 |
Table 2. DPS and LDPS numbered sequences
|
DPS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LDPS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
2 |
3 |
2 |
2 |
0 |
0 |
0 |
9 |
22 |
|
1=σ(8) |
1 |
2 |
3 |
2 |
0 |
0 |
0 |
8 |
22 |
|
2 |
2 |
3 |
3 |
0 |
0 |
0 |
0 |
8 |
17 |
|
2=σ(1) |
2 |
3 |
2 |
2 |
0 |
0 |
0 |
9 |
22 |
|
3 |
3 |
3 |
1 |
0 |
0 |
0 |
0 |
7 |
12 |
|
3=σ(2) |
2 |
3 |
3 |
0 |
0 |
0 |
0 |
8 |
17 |
|
4 |
3 |
4 |
0 |
0 |
0 |
0 |
0 |
7 |
11 |
|
4=σ(5) |
2 |
4 |
1 |
0 |
0 |
0 |
0 |
7 |
13 |
|
5 |
2 |
4 |
1 |
0 |
0 |
0 |
0 |
7 |
13 |
|
5=σ(6) |
2 |
4 |
2 |
0 |
0 |
0 |
0 |
8 |
16 |
|
6 |
2 |
4 |
2 |
0 |
0 |
0 |
0 |
8 |
16 |
|
6=σ(3) |
3 |
3 |
1 |
0 |
0 |
0 |
0 |
7 |
12 |
|
7 |
3 |
3 |
2 |
0 |
0 |
0 |
0 |
8 |
15 |
|
7=σ(7) |
3 |
3 |
2 |
0 |
0 |
0 |
0 |
8 |
15 |
|
8 |
1 |
2 |
3 |
2 |
0 |
0 |
0 |
8 |
22 |
|
8=σ(4) |
3 |
4 |
0 |
0 |
0 |
0 |
0 |
7 |
11 |
Table 3. ΔPS and LΔPS numbered sequences
|
ΔPS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LΔPS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
0 |
0 |
0 |
1 |
4 |
2 |
1 |
8 |
43 |
|
1=σ(1) |
0 |
0 |
0 |
1 |
4 |
2 |
1 |
8 |
43 |
|
2 |
0 |
0 |
0 |
1 |
4 |
4 |
0 |
9 |
48 |
|
2=σ(6) |
0 |
0 |
0 |
1 |
4 |
3 |
0 |
8 |
42 |
|
3 |
0 |
0 |
2 |
4 |
3 |
0 |
0 |
9 |
37 |
|
3=σ(2) |
0 |
0 |
0 |
1 |
4 |
4 |
0 |
9 |
48 |
|
4 |
0 |
0 |
2 |
4 |
2 |
1 |
0 |
9 |
38 |
|
4=σ(5) |
0 |
0 |
0 |
4 |
0 |
4 |
1 |
9 |
47 |
|
5 |
0 |
0 |
0 |
4 |
0 |
4 |
1 |
9 |
47 |
|
5=σ(4) |
0 |
0 |
2 |
4 |
2 |
1 |
0 |
9 |
38 |
|
6 |
0 |
0 |
0 |
1 |
4 |
3 |
0 |
8 |
42 |
|
6=σ(3) |
0 |
0 |
2 |
4 |
3 |
0 |
0 |
9 |
37 |
|
7 |
1 |
0 |
0 |
1 |
4 |
2 |
0 |
8 |
37 |
|
7=σ(8) |
1 |
0 |
0 |
0 |
1 |
4 |
2 |
8 |
44 |
|
8 |
1 |
0 |
0 |
0 |
1 |
4 |
2 |
8 |
44 |
|
8=σ(7) |
1 |
0 |
0 |
1 |
4 |
2 |
0 |
8 |
37 |
Table 4. ATS and LATS numbered sequences
|
ATS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LATS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
0 |
0 |
0 |
3 |
4 |
1 |
1 |
9 |
45 |
|
1=σ(8) |
0 |
0 |
0 |
0 |
1 |
3 |
2 |
6 |
37 |
|
2 |
0 |
0 |
1 |
1 |
3 |
4 |
0 |
9 |
46 |
|
2=σ(1) |
0 |
0 |
0 |
3 |
4 |
1 |
1 |
9 |
45 |
|
3 |
0 |
0 |
1 |
5 |
3 |
0 |
0 |
9 |
38 |
|
3=σ(5) |
0 |
0 |
1 |
1 |
0 |
4 |
1 |
7 |
38 |
|
4 |
0 |
1 |
0 |
3 |
2 |
1 |
0 |
7 |
30 |
|
4=σ(2) |
0 |
0 |
1 |
1 |
3 |
4 |
0 |
9 |
46 |
|
5 |
0 |
0 |
1 |
1 |
0 |
4 |
1 |
7 |
38 |
|
5=σ(3) |
0 |
0 |
1 |
5 |
3 |
0 |
0 |
9 |
38 |
|
6 |
0 |
1 |
0 |
0 |
3 |
3 |
0 |
7 |
35 |
|
6=σ(6) |
0 |
1 |
0 |
0 |
3 |
3 |
0 |
7 |
35 |
|
7 |
1 |
0 |
0 |
1 |
3 |
2 |
0 |
7 |
32 |
|
7=σ(4) |
0 |
1 |
0 |
3 |
2 |
1 |
0 |
7 |
30 |
|
8 |
0 |
0 |
0 |
0 |
1 |
3 |
2 |
6 |
37 |
|
8=σ(7) |
1 |
0 |
0 |
1 |
3 |
2 |
0 |
7 |
32 |
Table 5. DTS and LDTS numbered sequences
|
DTS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LDTS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
0 |
0 |
0 |
2 |
0 |
0 |
0 |
2 |
8 |
|
1=σ(8) |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
2 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
3 |
|
2=σ(1) |
0 |
0 |
0 |
2 |
0 |
0 |
0 |
2 |
8 |
|
3 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
3 |
|
3=σ(3) |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
3 |
|
4 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
|
4=σ(5) |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
3 |
|
5 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
3 |
|
5=σ(2) |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
3 |
|
6 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
|
6=σ(4) |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
|
7 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
|
7=σ(6) |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
|
8 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
|
8=σ(7) |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
1 |
Table 6. ΔTS and LΔTS numbered sequences
ΔTS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LΔTS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
0 |
0 |
0 |
0 |
2 |
1 |
1 |
4 |
23 |
|
1=σ(8) |
0 |
0 |
0 |
0 |
0 |
2 |
2 |
4 |
26 |
|
2 |
0 |
0 |
0 |
0 |
1 |
4 |
0 |
5 |
29 |
|
2=σ(5) |
0 |
0 |
0 |
0 |
0 |
3 |
1 |
4 |
25 |
|
3 |
0 |
0 |
0 |
2 |
3 |
0 |
0 |
5 |
23 |
|
3=σ(6) |
0 |
0 |
0 |
0 |
1 |
3 |
0 |
4 |
23 |
|
4 |
0 |
0 |
0 |
2 |
1 |
1 |
0 |
4 |
19 |
|
4=σ(2) |
0 |
0 |
0 |
0 |
1 |
4 |
0 |
5 |
29 |
|
5 |
0 |
0 |
0 |
0 |
0 |
3 |
1 |
4 |
25 |
|
5=σ(1) |
0 |
0 |
0 |
0 |
2 |
1 |
1 |
4 |
23 |
|
6 |
0 |
0 |
0 |
0 |
1 |
3 |
0 |
4 |
23 |
|
6=σ(4) |
0 |
0 |
0 |
2 |
1 |
1 |
0 |
4 |
19 |
|
7 |
1 |
0 |
0 |
0 |
2 |
2 |
0 |
5 |
23 |
|
7=σ(3) |
0 |
0 |
0 |
2 |
3 |
0 |
0 |
5 |
23 |
|
8 |
0 |
0 |
0 |
0 |
0 |
2 |
2 |
4 |
26 |
|
8=σ(7) |
1 |
0 |
0 |
0 |
2 |
2 |
0 |
5 |
23 |
Table 7. AVS and LAVS numbered sequences
|
AVS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LAVS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
3 |
6 |
7 |
8 |
8 |
8 |
8 |
48 |
212 |
|
1=σ(8) |
2 |
4 |
7 |
8 |
8 |
8 |
8 |
45 |
207 |
|
2 |
3 |
6 |
8 |
8 |
8 |
8 |
8 |
49 |
215 |
|
2=σ(1) |
3 |
6 |
7 |
8 |
8 |
8 |
8 |
48 |
212 |
|
3 |
4 |
7 |
8 |
8 |
8 |
8 |
8 |
51 |
218 |
|
3=σ(2) |
3 |
6 |
8 |
8 |
8 |
8 |
8 |
49 |
215 |
|
4 |
4 |
8 |
8 |
8 |
8 |
8 |
8 |
52 |
220 |
|
4=σ(5) |
3 |
7 |
8 |
8 |
8 |
8 |
8 |
50 |
217 |
|
5 |
3 |
7 |
8 |
8 |
8 |
8 |
8 |
50 |
217 |
|
5=σ(6) |
3 |
7 |
8 |
8 |
8 |
8 |
8 |
50 |
217 |
|
6 |
3 |
7 |
8 |
8 |
8 |
8 |
8 |
50 |
217 |
|
6=σ(3) |
4 |
7 |
8 |
8 |
8 |
8 |
8 |
51 |
218 |
|
7 |
4 |
7 |
8 |
8 |
8 |
8 |
8 |
51 |
218 |
|
7=σ(7) |
4 |
7 |
8 |
8 |
8 |
8 |
8 |
51 |
218 |
|
8 |
2 |
4 |
7 |
8 |
8 |
8 |
8 |
45 |
207 |
|
8=σ(4) |
4 |
8 |
8 |
8 |
8 |
8 |
8 |
52 |
220 |
Table 8. DVS and LDVS numbered sequences
|
DVS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LDVS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
2 |
3 |
1 |
1 |
0 |
0 |
0 |
7 |
15 |
|
1=σ(8) |
1 |
2 |
3 |
1 |
0 |
0 |
0 |
7 |
18 |
|
2 |
2 |
3 |
2 |
0 |
0 |
0 |
0 |
7 |
14 |
|
2=σ(1) |
2 |
3 |
1 |
1 |
0 |
0 |
0 |
7 |
15 |
|
3 |
3 |
3 |
1 |
0 |
0 |
0 |
0 |
7 |
12 |
|
3=σ(2) |
2 |
3 |
2 |
0 |
0 |
0 |
0 |
7 |
14 |
|
4 |
3 |
4 |
0 |
0 |
0 |
0 |
0 |
7 |
11 |
|
4=σ(5) |
2 |
4 |
1 |
0 |
0 |
0 |
0 |
7 |
13 |
|
5 |
2 |
4 |
1 |
0 |
0 |
0 |
0 |
7 |
13 |
|
5=σ(6) |
2 |
4 |
1 |
0 |
0 |
0 |
0 |
7 |
13 |
|
6 |
2 |
4 |
1 |
0 |
0 |
0 |
0 |
7 |
13 |
|
6=σ(3) |
3 |
3 |
1 |
0 |
0 |
0 |
0 |
7 |
12 |
|
7 |
3 |
3 |
1 |
0 |
0 |
0 |
0 |
7 |
12 |
|
7=σ(7) |
3 |
3 |
1 |
0 |
0 |
0 |
0 |
7 |
12 |
|
8 |
1 |
2 |
3 |
1 |
0 |
0 |
0 |
7 |
18 |
|
8=σ(4) |
3 |
4 |
0 |
0 |
0 |
0 |
0 |
7 |
11 |
Table 9. ΔVS and LΔVS numbered sequences
|
ΔVS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
LΔVS |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
ΣiSi,j |
ΣiSi,j∙i |
|
1 |
0 |
0 |
0 |
1 |
3 |
2 |
1 |
7 |
38 |
|
1=σ(1) |
0 |
0 |
0 |
1 |
3 |
2 |
1 |
7 |
38 |
|
2 |
0 |
0 |
0 |
1 |
3 |
3 |
0 |
7 |
37 |
|
2=σ(2) |
0 |
0 |
0 |
1 |
3 |
3 |
0 |
7 |
37 |
|
3 |
0 |
0 |
1 |
3 |
3 |
0 |
0 |
7 |
30 |
|
3=σ(6) |
0 |
0 |
0 |
1 |
3 |
3 |
0 |
7 |
37 |
|
4 |
0 |
0 |
1 |
3 |
2 |
1 |
0 |
7 |