Leading eigenvalues of adjacency matrices of star-like graphs with fixed numbers of vertices and edges

For a sequence of adjacency matrices, describing the unfolding of a network from the graph of a star, through graphs of a broom, to the graph of a link with constant vertices and edges, we show that the leading eigenvalue (the spectral radius) satisfies a simple algebraic equation. The equation allows easy numerical computation of the leading eigenvalue as well as a direct proof of its monotonicity in terms of the maximal degree of vertices. Subjects: Science; Bioscience; Epidemiology; Mathematics & Statistics; Applied Mathematics


Introduction
Our study of how the leading eigenvalue, or the spectral radius, of an adjacency matrix of a network varies when the structure of the network changes is motivated by recent interests of research in how an infectious disease spreads over a network (Brauer, 2008;Diekmann & Heesterbeek, 2000;Newman, 2002Newman, , 2010. M.E.J. Newman stated in the introduction to Chapter 16 of (Newman, 2010): "once we have measured and quantified the structure of a network, how do we turn the results into

PUBLIC INTEREST STATEMENT
In a network of agents susceptible to an infectious disease, assume that we know all details of the structure of the network: whether and how different agents are connected to each other. How can we predict whether the infectious disease will cause an epidemic if one or more agents are infected? How can we design policies to alter the structure of the network in order to reduce the speed at which the disease spreads? Similar problems arise in many other networks. These problems can often be formulated as a mathematical problem of understanding how certain global quantities such as the leading eigenvalue of a nonnegative matrix depends on its entries. In a simple situation when a network in the shape of a star unfolding into a broom and eventually forming a link, we give a complete description of the leading eigenvalue in terms of the network structure.
predictions or conclusions about how the overall system will behave? Unfortunately, progress in this area has been far slower than progress on characterizing structure … ". Imagine that we have a network of n agents susceptible to an infectious disease just discovered in the network. Assume that 0 β ij 1 is the transmission rate from agent i to agent j and 0 δ i 1 is the recovery rate for agent i. Then, there is a close relation between the leading eigenvalue of the matrix A ¼ ða ij Þ, where a ij ¼ β ij ; iÞj; a ii ¼ δ i , and the essential parameters of the infectious disease: the reproductive number R 0 , the speed at which the infectious disease can spread in the network, and the rate at which the disease is eradicated from the network (Diekmann & Heesterbeek, 2000;Gatto, Mari, & Rinaldo, 2013;Jiang, 2016;Kostova, 2009). When the transmission rates and the recovery rates are independent of individual agent, the study of the leading eigenvalue of the matrix A on its entries is reduced to determining how the leading eigenvalue of an adjacency matrix depends on the topology of the network. There are abundant results in graph theory on this subject, e.g. (Brouwer & Haemers, 2010;Brualdi & Hoffman, 1985;Bunimovich & Webb, 2014;Li & Feng, 1979;Liu, Lu, & Tian, 2004;Patuzzi, de Freitas, & Del-Vecchio, 2014). Many address the dependence of the leading eigenvalue on the underlying undirected connected graph. However, the underlying structure of the graph is usually not motivated by actual problems in applications nor dynamic. In this short paper, our interest is restricted to the study of the follow particular problem: assuming that the structure of a graph of n agents is evolving along a well-defined path, how does its leading eigenvalue change? We solve this problem completely in the simple case where the graph unfolds from a star to a link while both numbers of vertices and edges remain constant. The leading eigenvalue can be uniquely determined by solving a quite simple algebraic equation. As a consequence, we obtain its asymptotic behavior as the size of the network increases and also a direct proof of its monotonicity in terms of the maximal degree of vertices.
The adjacency matrix A ¼ ða ij Þ of any graph is a square matrix with entries either 0 or 1. An entry a ij of an Adjacency matrix is 1 if and only if two vertices v i and v j are connected by an edge. When a graph is undirected, its adjacency matrix is a symmetric nonnegative matrix. By the Perron-Frobenius theorem, matrix A has an eigenvalue λ>0, the leading eigenvalue, which is simple and greater than the magnitude of any other eigenvalues of A (Berman & Plemmons, 1994;Brouwer & Haemers, 2010).
For k ¼ 1; 2; Á Á Á ; n À 1, let A n ðkÞ denote the adjacency matrix of the graph G k and let λ n ðkÞ be the leading eigenvalue of A n ðkÞ. Let m ¼ n À k À 1. We note that the diameter of the graph G k is exactly k and the unique maximal degree of vertices of G k is n À k þ 1 ¼ m þ 2.
Theorem 1. The leading eigenvalue of A n ðkÞ, λ n ðkÞ, decreases in k, k ¼ 2; Á Á Á ; n À 1. For n ! 7 and 1 k n À 4, (1) where x n ðkÞ 2 ð1; n À k À 1Þ is the unique solution to the equation Since the one-variable function t þ 1 t is an increasing function on the interval ½1; 1Þ, the monotonicity of λ n ðkÞ in k follows from the monotonicity of x n ðkÞ once the relation (1) is established. A direct estimation of the unique solution x n ðmÞ>1 to the equation x nÀm ¼ mxÀ1 mÀx leads to the following asymptotic behavior of x n ðmÞ when n approaches infinite.
Corollary 1. For each fixed m ! 3, lim n!1 x n ðmÞ ¼ m: Remarks 1. The monotonicity of λ n ðkÞ in k is known. It follows from a theorem in (Li & Feng, 1979) and also from Corollary 2.2 in (Patuzzi et al., 2014). Showing that the leading eigenvalue λ n ðkÞ satisfying the equations (1) and (2) is new. The equations give not only an alternative direct proof of the monotonicity of λ n ðkÞ in k, but also a way to estimate the rate of change of λ n ðkÞ in k. 2. This formulation of the leading eigenvalue is reminiscent of another interesting result concerning the limit points of all leading eigenvalues of trees with λ<3 (Hoffman, 1972). 3. The graph G k is called a broom in (Del-Vecchio, Gutman, Trevisan, & Vinagre, 2009).
The rest of the paper is devoted to the proof the main theorem. In Section 3.1, using the standard cofactor expansion, we first show the leading eigenvalue λ n ðkÞ of the adjacency matrix A n ðkÞ satisfies a recursive relation of characteristic polynomials of the tri-diagonal adjacency matrices. Solving this recursive relation, we obtain a representation of the eigenvalue λ n ðkÞ in terms of the unique solution of a simple algebraic equation. The monotonicity of λ n ðkÞ in terms of n follows easily. A detailed proof of the monotonicity of λ n ðkÞ in k via the implicit differentiation is in Section 3.2.

Proofs
For k ¼ 1; 2; Á Á Á ; n À 1, let P n ðkÞ denote the characteristic polynomial of the adjacency matrix A n ðkÞ : P n ðkÞ ¼ detðλI À A n ðkÞÞ. Let Q k ¼ detðλI À A k ðk À 1ÞÞ denote the characteristic polynomial of the tri-diagonal adjacency matrix A k ðk À 1Þ. We can directly verify via cofactor expansion that P n ðkÞ and Q k possess the following properties. The second recursive relation is obtained by expanding the determinant along the last column first and then the last row. Iterating the recursive relation, we obtain the third equation. The details of the proof of these recursive relations are omitted since the verification of them is straightforward. See also (Del-Vecchio et al., 2009;Kulkarni, Schmidt, & Tsui, 1999).
Lemma 1 When m ! 3, the leading eigenvalue λ n ðkÞ has the properties λ n ðkÞ>2 and λ n ðkÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi x n ðkÞ p þ 1 ffiffiffiffiffiffiffiffi xnðkÞ p , where x n ðkÞ 2 ð1; mÞ is the unique real solution to the equation Proof. We consider the recursive relation (1) in Proposition 1, The equation has a unique real solution for b if and only if λ ! 2. Otherwise, b is a complex number. We have and thus, Iterate Equation (6) and notice that Q 1 ¼ λ and Further iterate the last recursive relation, we have for i ¼ 1; 2; Á Á Á , Thus, when i ¼ k À 2, Sum up the geometric progression we have, for k ! 2, We now solve the equation Multiply both sides by b and simplify. We have ððn À kÞ À b 2 À 1Þb 2kþ2 ¼ ðn À kÞb 2 À b 2 À 1: Or, b 2kþ2 ¼ ðn À k À 1Þb 2 À 1 ðn À k À 1Þ À b 2 : Let m ¼ n À k À 1 and x ¼ b 2 . So we have We conclude that λQ k À ðn À kÞQ kÀ1 ¼ 0 has a real positive solution λ>2 if and only if the equation x nÀm ¼ mxÀ1 mÀx has a real positive solution in ð1; 1Þ. The existence and uniqueness of the solution to the equation x nÀm ¼ mxÀ1 mÀx in the interval ð1; 1Þ are proved in the next lemma.
Lemma 2. For m ! 3 and n>m; n ! 7, the equation x nÀm ¼ mxÀ1 mÀx has a unique solution x ¼ x n ðmÞ 2 ð1; mÞ and x n ðmÞ increases in n.

3.2.
Monotonicity of x n ðmÞ in m: m ! 3 The characterization of the leading eigenvalue of the adjacency matrix A n ðkÞ provides a direct way to prove its monotonicity in m. Let x n ðmÞ denote the unique solution to the equation Proposition 2. When n À m ! 3 and m ! 3, x n ðmÞ<x n ðm þ 1Þ<m.