Let Σ be Laurent phenomenon (LP) seed of rank n, A(Σ), U (Σ) and L(Σ) be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively.We prove that each seed of A(Σ) is uniquely defined by its cluster, and any two seeds of A(Σ) with n − 1 common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that U (Σ) is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of Σ. Besides, we obtain the standard monomial bases of L(Σ). We also prove that U (Σ) coincides with L(Σ) under certain conditions.

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Let Q (resp. Q) be an extended exchange (resp. exchange) cluster quiver of finite mutation type. We introduce the distribution set of the number of arrows for M ut[ Q] (resp. M ut[Q]), give the maximum and minimum numbers of the distribution set and establish the existence of an extended complete walk (resp. a complete walk). As a consequence, we prove that the distribution set for M ut [ Q] (resp. M ut[Q]) is continuous except the exceptional cases.In case of cluster quivers Q inf of infinite mutation type, the number of arrows does not present a continuous distribution. Besides, we show that the maximal number of arrows of quivers inis infinite if and only if the maximal number of arrows between any two vertices of a quiver in M ut[Q inf ] is infinite. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Quivers of finite mutation type 3 2.2. Cluster quivers arising from surfaces 4 3. The proof of Theorem 1.3 with respect to exchange cluster quivers 6 3.1. Two lemmas 6 3.2. The maximum and minimum numbers of arrows of Q arising from surfaces 8 3.3. The existence of a complete walk 12 4. The proof of Theorem 1.3 with respect to extended exchange cluster quivers 20 4.1. Two lemmas 20 4.2. The maximum and minimum numbers of arrows of Q arising from surfaces 21 4.3. The existence of an extended complete walk 24 5. The number of arrows of cluster quivers of infinite mutation type 27 5.1. Distribution of the numbers of arrows 27 5.2. The number of arrows between two vertices 27 6. Appendix 30 References 34 Date: version of May 25, 2021.

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<abstract><p>Let $ \Sigma $ be a Laurent phenomenon (LP) seed of rank $ n $, $ \mathcal{A}(\Sigma) $, $ \mathcal{U}(\Sigma) $, and $ \mathcal{L}(\Sigma) $ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $ \mathcal{A}(\Sigma) $ is uniquely defined by its cluster and any two seeds of $ \mathcal{A}(\Sigma) $ with $ n-1 $ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $ \mathcal{U}(\Sigma) $ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $ \Sigma $. Besides, we obtain the standard monomial bases of $ \mathcal{L}(\Sigma) $. We also prove that $ \mathcal{U}(\Sigma) $ coincides with $ \mathcal{L}(\Sigma) $ under certain conditions.</p></abstract>