On the Eventual Exponential Positivity of Some Tree Sign Patterns

An n× n matrix A is called eventually exponentially positive (EEP) if etA = ∞ ∑ k=0 tk Ak k! > 0 for all t ≥ t0, where t0 ≥ 0. A matrix whose entries belong to the set {+,−, 0} is called a sign pattern. An n× n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A ofA that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article,A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n ≤ 3 are identified. For the n× n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern T o n . Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of T o n . We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n, m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.


Introduction
An n × n real matrix A is called eventually positive (EP) if there exists a positive integer k 0 such that A k > 0 for all k ≥ k 0 , where the inequality is interpreted entrywise. An n × n real matrix A is called eventually exponentially positive (EEP) if there exists a nonnegative integer t 0 such that for all t ≥ t 0 ; see, e.g., [1]. EP matrices and EEP matrices have applications to the system of linear differential equationsẋ(t) = Ax(t) (A ∈ R n×n , x 0 = x(0) ∈ R n , t ≥ 0) whose solutions become positive at finite time and remain positive for all time thereafter; see e.g., [2]. A sign pattern is a matrix A = [α ij ] whose entries are in the set {+, −, 0}. An n × n real matrix A is called a realization of A if A has the same sign pattern as A. The qualitative class of sign pattern A is the set of all realizations of A, and is denoted by Q(A) . An n-by-n sign pattern B = [β ij ] is called a subpattern of A = [α ij ] (equivalently, A is called a superpattern of B) if β ij = 0 implies α ij = β ij for all i and j. If the subpattern B = A, then B is called a proper subpattern of A. A sign pattern A is called reducible if there exists a permutation matrix P such that where the square matrices A 11 and A 22 are nonvacuous. Ortherwise, A is called irreducible; see, e.g., [3][4][5] for more details and further terminology regarding sign patterns. An n × n sign pattern A is called potentially eventually positive (PEP), if there exists some A ∈ Q(A) such that A is EP. PEP sign patterns have been studied first in [6], where the PEP sign patterns of order at most 3 have been classified, and some sufficient conditions and some necessary conditions for an n × n (n ≥ 4) sign pattern to be PEP have been established. However, characterizing the n × n (n ≥ 4) PEP sign pattern is still a longstanding open problem.
An n × n sign pattern A is called potentially eventually exponentially positive (PEEP), if there exists some A ∈ Q(A) such that A is EEP. Some sufficient conditions and some necessary conditions for an n × n sign pattern to be PEEP have been established in [7]. However, there are still not any characterization about the PEEP sign patterns.
In this article, we investigate the potential eventual exponential positivity of some tree sign patterns by considering its minimality. Our work is organized as follows. In Section 2, we introduce the MPEEP sign patterns to classify the PEEP sign patterns, and some preliminary results for an n × n sign pattern A to to be EEP are established. All MPEEP sign patterns of small orders are also identified as the section's conclusion. In Section 3, all PEEP tridiagonal sign patterns are classified by identifying the unique MPEEP tridiagonal sign pattern T o n . In Section 4, the unique MPEEP star sign pattern S o n is identified and consequently, all PEEP star sign patterns are classified to be a superpattern of S o n . In Section 5, all PEEP double star sign patterns are classified by identifying the exactly one MPEEP double star sign pattern DS o n,m . Some discussions and conclusions are made in Section 6.

Terminologies and Preliminary Results
As a beginning, some necessary terminologies regarding graph theory are introduced below. For further details, see [1,3] and the references therein.
A square sign pattern A = [α ij ] is called combinatorially symmetric if α ij = 0 whenever α ji = 0. Let G(A) be the graph of order n with vertices 1, 2, . . ., n and an edge {i, j} joining the distinct vertices i and j if and only if α ij = 0. Then G(A) is called the graph of sign pattern A. A combinatorially symmetric sign pattern matrix A is called a tree sign pattern (respectively, star sign pattern, double star sign pattern) if G(A) is a tree (respectively, star, double star).
It is known from [7] that sign pattern A is PEEP if the positive part A + is irreducible. Since every superpattern of PEEP sign patterns is also PEEP, it is necessary to investigate the minimality of PEEP sign patterns. Consequently, the following terminology is proposed.

Definition 1.
Sign pattern A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and any proper subpattern of A is not PEEP.
Please note that sign pattern A is PEEP if and only if P T AP or P T A T P is PEEP, for any permutation pattern P. As a result, two sign patterns A and B are called equivalent if B = P T AP or B = P T A T P for some permutation pattern P. By considering the minimality of PEEP sign patterns, we obtain the following Proposition 1.

Proposition 1.
Let A be an n × n sign pattern. Then the following statements are equivalent: 1.
A T is MPEEP; 3. P T AP is MPEEP, where P is an n × n permutation pattern.
For an n × n real matrix A, let σ(A) be .
Recall that a real eigenvalue γ ∈ σ(A) (the spectrum of A) is called rightmost, if γ is simple, and Re(λ) < γ for all λ ∈ σ(A) (λ = γ); see [7] for details. The following Lemma 1 follows readily from the Theorem 3.3 in [2] and Proposition 1.6 in [7] Lemma 1. Let A be an n × n real matrix. Then the following statements are equivalent: 1.
A + αI is EP for some α ≥ 0, where I is an identity matrix of order n; 3.
A has a rightmost eigenvalue with positive right and left eigenvectors.
Following [7], let A D(+) (respectively, A D(0) and A D(−) ) be the sign pattern whose off-diagonal entries are the same as A and all the diagonal entries are + (respectively, 0 and −). Observation 1.4 in [7] is cited here as Lemma 2 for the sake of convenience.

Lemma 2.
If an n × n sign pattern A is PEEP, then A D(+) is PEP.
It is known from [7] that the converse of Lemma 2 is not true. But for the sign patterns with only − on the diagonal, the converse is true.  [7]. However, Next, we turn to identify all the MPEEP sign patterns of orders n ≤ 3. It is clear that a 1 × 1 sign pattern A is a MPEEP sign pattern if and only if A = (0). It is known that all 2 × 2 and 3 × 3 PEEP sign patterns were classified in [7]. We cite these results below for the readers' convenience to identify all the 2 × 2 and 3 × 3 MPEEP sign patterns.

Then there exists an EP matrix realization
In the following two propositions, we identify all MPEEP sign patterns of orders 2 and 3, respectively.
Proof. Sign pattern A = 0 + + 0 is PEEP for its positive part is irreducible. Moreover, it is clear that no proper subpattern of A is irreducible and PEEP. It follows that A is MPEEP. For the converse, assume that A is MPEEP. Then A is of the form in Lemma 3.
in the proof of A 3 . If (A 4 ) 22 is changed to be 0, then the corresponding sign pattern has no − on the diagonal, no + in row 1, and thus cannot be PEEP. Therefore, A 4 is MPEEP.
For the necessity, suppose that A is a MPEEP sign pattern of order 3. Then A is equivalent to one of the sign patterns stated in Lemma 4. The necessity follows from the fact that sign patterns A 1 , A 2 , A 3 and A 4 are MPEEP.

MPEEP Tridiagonal Sign Patterns
In this section, we identify all MPEEP tridiagonal sign patterns and classify all PEEP tridiagonal sign patterns. Without loss of generality, let an n × n tridiagonal sign pattern where * denotes the nonzero entries, ? denotes one of 0, +, − and the entries unspecified in the sign pattern are all zeros. Throughout this section, let n ≥ 4. The following Lemma 5 is very important to identify the MPEEP sign patterns in the class of tridiagonal sign patterns.

Lemma 5.
If the tridiagonal sign pattern T n is PEEP, then all nonzero off-diagonal entries of T n are +.
Proof. Since tridiagonal sign pattern T n is PEEP, (T n ) D(+) is PEP by Lemma 2. By the Theorem 1 in [8], all nonzero off-diagonal entries of (T n ) D(+) are +. It follows that all nonzero off-diagonal entries of T n are +.
Now we identify all n × n MPEEP tridiagonal sign patterns.

Theorem 1. The sign pattern
is the unique (up to equivalence) MPEEP tridiagonal sign pattern.
Proof. T o n is PEEP follows from the fact that the positive part (T o n ) + is irreducible. every proper subpattern of T o n is not PEEP by Lemma 5. It follows that T o n is MPEEP. Now let T is an arbitrary n × n MPEEP tridiagonal sign pattern. Then all nonzero off-diagonal entries of T are + by Lemma 5. If T has at least one nonzero diagonal entry, then T must be a superpattern of T o n , and hence T is not MPEEP. Consequently, all diagonal entries of T must be 0.Therefore, T is equivalent to T o n .
We conclude this section by classifying the PEEP tridiagonal sign patterns which can be shown readily from Theorem 1.

Corollary 2.
Let T n be an n × n tridiagonal sign pattern. Then T n is PEEP if and only if T n is equivalent to a superpattern of T o n .

MPEEP Star Sign Patterns
In this section, we identify all MPEEP star sign patterns and classify all PEEP star sign patterns. Generally, the n-by-n ( n ≥ 4) star sign pattern To classify the PEEP star sign patterns, it is very necessary to investigate its nonzero off-diagonal entries. Lemma 6. If S n is PEEP, then all nonzero off-diagonal entries of S n are +.
Proof. (S n ) D(+) is PEP follows from the fact that star sign pattern S n is PEE and Lemma 2. By the Theorem 2.7 in [9], all nonzero off-diagonal entries of (S n ) D(+) are +. It follows that all nonzero off-diagonal entries of S n are +.
To proceed, we turn to identify all n × n MPEEP star sign patterns.

Theorem 2. The sign pattern
is the unique (up to equivalence) MPEEP star sign pattern.
Proof. Clearly, the positive part of S o n is irreducible. It follows that S o n is PEEP. By Lemma 6, every proper subpattern of S o n is not PEEP. Consequently, S o n is MPEEP. For the uniqueness, let S be an arbitrary n × n MPEEP star sign pattern. Then by Lemma 6, all nonzero offdiagonal entries of S are +. To complete the proof, it suffices to show the diagonal entries are 0. By a way of contradiction, let some of diagonal entries of S be nonzero. Then S is a superpattern of S o n , and hence S is not MPEEP; a contradiction. Therefore, all diagonal entries of S must be 0.
We conclude this section by classifying all n × n PEEP star sign patterns which can be shown directly from Theorem 2.
Corollary 3. An n × n star sign pattern S n is PEEP if and only if S n is equivalent to a superpattern of S o n .

MPEEP Double Star Sign Patterns
Following [10], let the (n + m) × (n + m) double star sign pattern To state clearly, let (ds) i,j be the (i, j)-th entry of double star sign pattern DS(n, m). Below is a necessary condition for an (n + m) × (n + m) double star sign pattern DS(n, m) to be PEEP.
is the unique (up to equivalence) MPEEP double star sign pattern.
Theorem 3 indicates that for the class of (n + m) × (n + m) double star sign patterns, there exists a unique sign pattern that is MPEEP, which classifies all PEEP double star sign patterns.

Discussions and Conclusions
We have introduced the MPEEP sign patterns, which is just a preliminary study. However, our results have indicated that identifying the MPEEP sign patterns is easier than identifying the PEEP sign patterns, and that an effective method to classify the PEEP sign patterns is identifying the MPEEP sign patterns. In fact, we have classified all the PEEP tridiagonal (respectively, star, double star) by identifying the unique MPEEP tridiagonal (respectively, star, double star) sign pattern, up to equivalence. Our results and method are of great significance for characterizing other tree sign patterns. Motivated by the obtained results, here we propose two conjectures for further studying.

Conjecture 1.
For the n × n tree sign pattern class, there exists a unique MPEEP tree sign pattern, and consequently the corresponding PEEP tree sign patterns are exactly its superpatterns.

Conjecture 2.
If an n × n tree sign pattern A is MPEEP, then A must have 2(n − 1) positive entries.
Please note that the minimum number of positive entries in an n × n PEEP sign pattern is unknown. What we have known is just that the upper bound for the minimum number of positive entries is n from [7].
Author Contributions: B.-L.Y. is responsible for providing methods, proofs, analysis processes and checking the final manuscript. Z.L. is responsible for making a detailed analysis and correcting grammatical errors, and S.X. is responsible for writing and editing the orignial draft. All authors have read and agreed to the published version of the manuscript.