1 On the monotonicity of the number of positive 2 entries in nonnegative five element matrix powers 3 M. Rodriguez Galvan, V. Ponomarenko, and J. Salvadore Sabio 4 April 10, 2021 5 Abstract 6 Let A be an m×m square matrix with nonnegative entries and let F (A) denote the number of 7 positive entries in A. We consider the adjacency matrix A with a corresponding digraph with 8 m vertices. F (A) corresponds to the number of directed edges in the corresponding digraph. n 1 9 We consider conditions on A to make the sequence fF (A )gn=1 monotonic. Monotonicity is 2 10 known for F (A) ≤ 4 (except for 3 non-monotonic cases) or F (A) ≥ m − 2m + 2; we extend 11 this to F (A) = 5. 12 13 Keywords: nonnegative matrix; power; monotonicity; directed graph; adjacency matrix 14 15 Mathematical Subject Classification 2020: 15B34, 15B48, 05C20 16 1 Introduction 17 Nonnegative matrices are matrices with nonnegative real entries. Nonnegative matrices are valuable 18 to study as they can be applied to fields such as probability, economics, and combinatorics (see [1]). 19 We define F to be a function from the nonnegative square matrices to the integers that counts the 20 number of positive entries in nonnegative square matrices. Then for any nonnegative matrix A, we n 1 21 can classify the sequence fF (A )gn=1 as non-monotonic, monotonically increasing, monotonically 22 decreasing, or constant. 23 It is clear to see that the value of each positive matrix value does 24 not give any greater insight into the question of monotonicity. Thus, + 0 1 × 0 1 25 we can define our matrices to be of Boolean propositions as defined 0 0 1 0 0 0 26 in [4]. These propositions can be one of two elements, unity and zero, 1 1 1 1 0 1 27 with the operations presented to the right for reference. 28 For brevity, we can call these square matrices of Boolean proposi- 29 tions 0-1 matrices. There is a correspondence between a directed graph and a 0-1 matrix, known 30 as an "adjacency matrix". It is common to observe adjacency matrices of digraphs (see [8]), with 31 adjacency matrices already conforming to the Boolean propositions seen in 0-1 matrices. Adjacency 32 matrices have many applications, examples including when studying strongly regular graphs and 1 33 two-graphs (see [7]). Another application of the adjacency matrices we are studying is that adja- 34 cency matrices of strongly connected graphs are irreducible, and so the Perron-Frobenius Theorem 35 can be related to these matrices (see [3]). th th 36 An adjacency matrix has a 1 in its i row and j column if 37 0 1 there is a directed edge from vertex i to vertex j. Since we are 1 o 2 0 0 1 0 38 @ O B1 0 0 1C examining adjacency matrices that are 0-1 matrices, we need 39 B C not consider digraphs with repeated edges from one vertex to @0 1 1 1A 40 3 / 4 0 1 0 0 another. To the left is an example of this correspondence. [ k 41 We note that the (i; j) entry of an adjacency matrix A shows 42 whether or not there is at least 1 directed path of length k from 43 vertex i to vertex j (if more than one path of length k exists, the adjacency matrix entry is unity k 44 regardless). So, we have that F (A ) is equal to the number of edges in the digraph corresponding to k 45 the adjacency matrix A . Then, we can instead observe the number of directed edges in a digraph k 46 composed with itself k times instead of directly computing A and counting the number of positive 47 entries. Thus, we only need examine directed graphs with 5 edges (no repeated edges) and verify 48 whether the number of edges as the digraph is composed with itself repeatedly is monotonic or not. 49 We note that adjacency matrices of undirected graphs are symmetric, but as our goal is to study 50 nonnegative square matrices, it is more useful to study the adjacency matrices of directed graphs 51 that may or may not be symmetric. 52 In a brief digression, we note how if we have a digraph that has disjoint parts, then we can k 53 observe this corresponds to a block diagonal adjacency matrix, denote it B. As we find B for any 54 k 2 N, we note that the entries in each block do not affect the entries in another block. As such, 55 each of the digraph's disjoint parts, which correspond to a block in B, will never develop edges k 56 that connect the disjoint parts for any B . This means if we have that each of the disjoint parts of 57 the digraph, corresponding to blocks, have a monotonically increasing number of edges, then the 58 adjacency matrix B will be monotonically increasing. We reach a similar conclusion if the parts of 59 the digraphs all have a monotonically decreasing number of edges. So if we have a digraph case with 60 disjoint parts that are all monotonically increasing/decreasing we can easily determine the whole 61 digraph to be monotonic and not include it in our list of cases in this paper's Section 3. However, 62 if we have a digraph case with disjoint parts that have some being monotonically increasing and 63 others being monotonically decreasing, then we include these cases in our work as we can reach no 64 such conclusion. 65 In [2] and [9]; Xie, Brower, and Pono- 2 3 66 marenko proved that for any m × m 0-1 matrix a / b a / b a / b 2 O o O O ^ 67 A, if F (A) ≤ 4 or F (A) ≥ m − 2m + 2 then n 1 6 7 68 the sequence fF (A )g is monotonic, except 4 5 n=1 69 for 3 non-monotonic cases. To the right are the c c o d c d 70 only non-monotonic cases. 71 Our results allow us to conclude that all 0-1 matrices A, with F (A) = 5, are monotonic, except 72 for the following cases shown below. Dotted edges demonstrate that some of these non-monotonic 73 cases for F (A) = 5 have subsets that form a non-monotonic case from previous work. 74 2 2 a / co e a o c e a / co e a o c / e a / b e 3 O O e O _ O O o @ 6 7 6 7 6 7 6 b o d b / d b d b d c d 7 6 7 6 a o c / e a / c e a / c e a / c e a / b 7 6 O O : O E O O o 7 6 7 6 7 6 7 6 b d f 7 6 b / d b o d b o d o c d 7 6 / 7 6 a / c5 e a i / c e a / c e a / c e a o b 7 6 O _ O _ O _ Y O ^ O 7 6 7 75 6 7 6 7 6 b d b d b d b d f c d 7 6 7 6 a / b / d a / b / c a o b o c a / b a / b 7 6 O ^ Y @ Y ^ O O ^o 7 6 7 6 7 6 7 6 Ð Ð Ð 7 6 c e d e d e c d c d 7 6 7 6 a / b a / b ao b / c a / b o c 7 6 O ^ ^ ^ A O ^ 7 6 7 4 5 Ñ d c e d c e d e f d e f 76 In the second section, we establish some important terminology and theorems that will be useful 77 in proving the monotonicity of the number of positive entries in nonnegative five element matrix 78 powers. 79 2 Theorems 80 To begin, consider the digraph and corresponding adjacency matrix 81 1 / 2 0 1 A to the left. We have F (A) = 3, and through direct calculation we ; 0 1 0 k 1 T 82 have that the sequence fF (A )gk=2 = 2. Now, let us consider A . A @0 0 0A T 83 Similarly, we have through direct calculation that F (A ) = 3 and 0 1 1 3 p T k 1 84 fF ((A ) )gk=2 = 2. We also observe that graphically, the digraph 0 1 85 1o 2 0 0 0 corresponding to A has the direction of its directed edges switched T T 86 A @1 0 1A when finding the digraph corresponding to A . With this in mind, k T k 87 Ó 0 0 1 we want to prove that for any k 2 , F (A ) = F ((A ) ) when con- 3 p N 88 sidering adjacency matrices. This would mean that if we can prove, 89 either through calculation or a theorem later in this paper, that a k 1 90 certain kind of digraph corresponding to an adjacency matrix has the sequence fF (A )gk=1 being 91 monotonic, then we can say the digraphs where the direction of the directed edges are reversed are 92 also monotonic. The following theorem proves just that. k T k 93 Theorem 2.1. Let A be a square 0-1 matrix. Then 8k 2 N, F (A ) = F ((A ) ).