SECTIONS NOTES ON GRAPH THEORY NOTATION AND ITS USE IN THE STUDY OF SPARSE SYMMETRIC MATRICES


 Ethelbert Cook
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1 SECIONS.5.6 NOES ON GRPH HEORY NOION ND IS USE IN HE SUDY OF SPRSE SYMMERIC MRICES graph G ( X, E) consists of a finite set of nodes or vertices X and edges E. EXMPLE : road map of part of British Columbia Kamloops Whistler Merrit Vancouver Hope Princeton he information contained in this map can be represented by a graph with N 6 vertices. X { Van, Whi, Kam, Hop, Pri, Mer } E { (Van, Whi), (Whi, Kam), (Hop, Van), (Kam, Mer), (Hop, Pri), (Mer, Hop) } Note that E is a set of unordered pairs; that is, (Van, Whi) is the same edge as (Whi, Van). Usually a graph is represented as follows (rather than by listing the sets X and E): Van Hop Mer Pri Kam Whi 3
2 Note that although the above graph looks quite different from the map of B.C., all of the information contained in the sets X and E is retained. he above is an example of an unordered (or unlabelled) graph. n ordering (or labelling) α is a mapping of the integers {, 2,, N} onto X. For example, if α is the mapping Van 2 Hop 3 Mer 4 Pri 5 Kam 6 Whi then the above unordered graph becomes the ordered graph in Figure 3.. : he relationship between graphs on N nodes and N N symmetric matrices: an N N symmetric matrix has an associated ordered graph with node set X {, 2,, N } and edge set E such that ( i, j) ( j, i) E if and only if a a and i j. ij ji s we will be interested only in positive definite matrices, which always have all diagonal entries nonzero, we will put nonzeros in all positions on the main diagonal. hus, as in Figure 3.., the matrix associated with the above graph has nonzeros in positions indicated by an as follows: 32
3 33 (Note in the George/Liu notes, the diagonal entries are denoted by circled integers, rather than.) Let I P be an N N permutation matrix. hen PP is a symmetric reordering of the rows and columns of. For example, in Figure 3..2, P and (using the above matrix ) PP and the associated graph for this matrix is
4 How do you determine the permutation matrix P such that for the original matrix above and its associated graph, the graph associated with PP is as above? he nonzeros in P can be determined as follows: Mapping of the nodes Nonzeros in P 2 (2, ) 2 4 (4, 2) 3 3 (3, 3) 4 6 (6, 4) 5 (, 5) 6 5 (5, 6) he unlabelled (or unordered) graphs of and PP are the same they represent the structure of or the equivalence class of all matrices PP where P is any N N permutation matrix. he ordered graphs are associated with PP for different permutation matrices P. he problem of finding a good permutation matrix for (with respect to some sparse matrix problem) is equivalent to finding a good ordering (or labeling) of the graph of. ERMINOLGY wo nodes x and y are adjacent in a graph G if the nodes x and y are said to be neighbors. he adjacent set of a node y in G is ( x, y) ( y, x) E. In this case, dj(y) { x X : ( x, y) E } he degree of a node y is dj (y)., the cardinality of the set dj(y). 34
5 (simple) path from node x to node y of length l in G is an ordered set of l + distinct nodes ( v, v2, K, v l + ) such that vi+ dj( vi ), for i, 2, K, l with v x and v l y. + graph G is connected if every pair of distinct nodes is joined by at least one path. Otherwise, G is disconnected and consists of two or more connected components. disconnected graph with two connected components: Relationship with matrices: the graph of a matrix is disconnected if and only if there exists a permutation matrix P such that PP 22 where and 22 are square (nonempty) submatrices. Such a matrix block diagonal matrix. PP is called a EXMPLE Suppose that a matrix has the following zero/nonzero structure:. he graph of is 35
6 With P, we obtain PP, which is a block diagonal matrix. he graph of PP is Definition symmetric matrix is reducible if there exists a permutation matrix P such that 22 PP, where 22 and are square (nonempty) submatrices. If such a permutation matrix P does not exist, then is irreducible. HEOREM symmetric matrix is reducible if and only if its associated graph is disconnected. symmetric matrix is irreducible if and only if its associated graph is connected.
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