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Number Theory and Graph Theory Chapter 9 Additional Topics

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Number Theory and Graph Theory Chapter 9 Additional Topics 1 Number Theory and Graph Theory Chapter 9 Additional topics By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: [email protected] 2 Module-1: Nonnegative matrices Objectives • Reducible and irreducible matrices • Statements of Perron and Perron Frobenius theorems. • Primitive matrices • Exponent of a primitive matrix A real matrix A having all its entries positive is called a positive matrix. We denote positive matrix by A > 0. Similarly, if each entry of a real matrix is non-negative, then we denote it as A ≥ 0 and call it as nonnegative matrix. Similarly, we can define a positive or nonnegative vector. A square matrix A of order n ≥ 2; is called reducible if there is a permutation matrix P such that 2 3 BC PT AP = 4 5; where B and D are square submatrices. Otherwise A is called irreducible. 0 D Definition 1. Let X be a directed graph and let B be its adjacency matrix. Then, X is said to be associated with a nonnegative matrix C if Ci j 6= 0 , Bi j 6= 0: We denoted the graph X associated with C as XC and its adjacency matrix by Ad(XC), i.e., Ad(XC) = B: If C is symmetric, then XC is undirected or equivalently Ad(XC) is symmetric. Note that a digraph can be associated with more than one matrix. For example, a complete graph with a self loop at each vertex is associated with every positive matrix, i.e., Ad(XC) = J whenever C > 0: 2 3 2 3 0 1 0 0 0 1 0 0 6 7 6 7 6 7 6 7 60 0 1 07 60 0 2 07 Example 2. Let A = 6 7 and B = 6 7, then the digraphs XA and XB are same. 6 7 6 7 61 0 0 17 67 0 0 37 4 5 4 5 1 0 0 1 4 0 0 5 3 3 4 1 2 XA or XB We will use the following result quite frequently. where is the proof of the next few results. Theorem 3. Let A = (ai j) be a nonnegative matrix. Then the following are equivalent. 1. A is irreducible. (k) k 2. For each (i; j), there exists an integer k such that ai j = (A )i j > 0. 3. XA is strongly connected. Theorem 4. Let A be an irreducible matrix of order n ≥ 2 and y be a nonnegative vector with exactly k positive entries, for some k, 1 ≤ k ≤ (n − 1). Then, (I + A)y has more than k positive entries. Corollary 5. A nonnegative matrix A is irreducible if and only if (I + A)n−1 > 0. Recall that for two n × n matrices A and B, A ≥ B if A − B ≥ 0. Lemma 6. Let A be an irreducible matrix and B ≥ A. Then, B is also an irreducible matrix. Proof. Let XA be the directed graph associated with the matrix A. Then, XA is strongly connected and hence XB is also strongly connected as Ad(XB) = Ad(XA) + D, where D a 0;1-matrix. Definition 7. Let A 2 Mn(C). Then, the spectral radius of A is denoted as r(A) and is define as r(A) = max fjlij : li is an eigenvalue ofAg: 1≤i≤n 4 2 3 −1 1 Example 8. Let A = 4 5, then r(A) = 1. 0 −1 The following theorem is due to Perron proved in 1907, called the Perron’s theorem. Theorem 9. (Perron’s theorem) Let A 2 Mn(R) and r(A) = jrj, s(A) = fl1;l2;:::;lsg, the set of all distinct eigenvalues of A. If A > 0 then the following statements are true. 1. r > 0. 2. r 2 s(A) (r is called the Perron Value). 3. r is a simple eigenvalue of A: T 4. There exists an eigenvector x > 0 such that Ax = rx. The unique vector, say P = (p1; p2;:::; pn) 2 n n R satisfying AP = rP, P > 0 and ∑ pi = 1 is called the Perron vector. i=1 5. r is the only eigenvalue on the spectral circle of A: Frobenius in 1912 extended Perron’s theorem of positive matrices to nonnegative irreducible matrices. It satisfies all the conditions of Person’s theorem except, spectral circle of irreducible matrix may contain more than one eigenvalue. Theorem 10. (Perron Frobenius theorem) Let A 2 Mn(R) and r(A) = jrj, s(A) = fl1;l2;:::;lsg, the set of all distinct eigenvalues of A. If A ≥ 0 is irreducible then the following statements are true. 1. r > 0. 2. r 2 s(A) (this r is also called the Perron Value of A). 3. r is a simple eigenvalue of A: T 4. There exists an eigenvector x > 0 such that Ax = rx. The unique vector, say P = (p1; p2;:::; pn) 2 n n R satisfying AP = rP, P > 0 and ∑ pi = 1 is called the Perron vector. i=1 5 Definition 11. Let A be an irreducible matrix with Perron value r, and suppose that A has h eigenvalues of modulus r. Then, the number h is called the index of imprimitivity of A, or simply the index of A. If h = 1, then the matrix A is said to be primitive, otherwise it is called imprimitive (cyclic). The following theorem establishes that the index of an irreducible matrix can be computed from its characteristic polynomial. n n n Theorem 12. Let A be an irreducible matrix with index h, and let f (x) = x +a1x 1 +a2x 2 +···+ n akx k , where n > n1 > n2> > ··· > nk, and at 6= 0, t = 1;2;:::;k be the characteristic polynomial of A. Then h = gcd(n − n1;n1 − n2;:::;nk−1 − nk): In terms of digraph, the above result can be stated as “if A is an irreducible matrix with index h and XA is its associated digraph, then h is equal to the greatest common divisor of lengths of all circuits in XA”. For example, the index of the adjacency matrix A of the following graph is 1 hence A is primitive. As each edge is a directed cycle of length 2, in the given graph, there are cycles of lengths 2;3 and 4: The following theorem helps in determining whether a given nonnegative matrix is primitive or not. Theorem 13. Let A be a nonnegative matrix, then A is primitive if and only if there exists a positive integer m such that Am > 0: 2 3 0 1 0 6 7 6 7 3 2 Example 14. The characteristic polynomial of A = 60 0 17 is x − x − x − 1. Hence clearly 4 5 1 1 1 6 2 3 2 3 1 1 1 0 1 0 6 7 6 7 3 6 7 6 7 h = 1: One can verify A = 61 2 27 > 0: The characteristic polynomial of B = 60 0 17. Note 4 5 4 5 2 3 4 1 0 0 that B is irreducible (as it is the adjacency matrix of directed cycle) and its eigenvalues lie on the unit circle as they are the cube roots of unity. Hence, its index of imprimitivity h = 3: In general, the index of imprimitivity of Wn is n: Let A 2 Mn(R) be primitive. Then, by previous theorem, there exists a positive integer m such m m that A > 0: So, SA = fm 2 N : A > 0g, is a non empty subset of N. By well ordering principle, SA has a least element say k. This k is called the exponent of A, and is denoted as exp(A). In terms of a graph, the exponent of a matrix A is the smallest positive integer k such that is a walk of length k in XA, for each vertex i and j. Example 15. • Any positive matrix has exponent 1: 2 • If A is the adjacency matrix of the complete graph Kn then exp(A) = 2, as A > 0: 2 3 2 3 2 3 0 1 0 0 0 1 1 1 1 6 7 6 7 6 7 6 7 2 6 7 3 6 7 • Let A=60 0 17 then, A = 61 1 17 and, A = 61 2 27 Hence, exp(A) = 3: 4 5 4 5 4 5 1 1 1 1 2 2 2 3 4 • Every primitive matrix is irreducible but the converse is not true. For example, the matrix Wn (the adjacency matrix of a directed cycle) is irreducible but not primitive. Proof. Since directed cycle is strongly connected hence Wn is irreducible. However, the n characteristic polynomial of Wn is x − 1, and hence all its roots have the same modulus 1. • If A is the adjacency matrix of a connected graph X, then A is irreducible. But A is primitive if and only if X is not bipartite. 7 Proof. Since any connected graph is strongly connected, its adjacency matrix is irreducible. 2k+1 Let A be the adjacency matrix of a bipartite graph. Then (A )ii = 0, for all k and all i, as there are no cycles of odd length. Hence, A is not primitive. Since in a connected graph every edge is a directed cycle of length 2. Consequently, A is primitive if X contains a cycle of odd length. Hence the result follows. • A graph/digraph is said to be primitive if its adjacency matrix is primitive. A graph is not primitive if and only if it is either disconnected or connected bipartite.
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