Abstract. In two recent papers, one by Friedland and Schneider, the other by Förster and Nagy, the authors used polynomial matrices to study the effect of graph expansions on the spectral radius of the adjacency matrix. Here it is shown that the notion of the elasticity of the entries of a nonnegative matrix coupled with the Variational Principle for Pressure from symbolic dynamics can be used to derive sharper bounds than existing estimates. This is achieved for weighted and unweighted graphs, and the case of equality is characterized. The work is within the framework of studying measured graphs where each edge is assigned a positive length as well as a weight.Key words. Graph expansions, Nonnegative matrices, Elasticity, Matrix polynomials.AMS subject classifications. 05C50, 15A48, 37B10, 92D25.
Introduction.In two recent papers Förster and Nagy [6] and Friedland and Schneider [7] consider the effect of an expansion of unweighted and weighted directed graphs, respectively, on the spectral radius of its adjacency matrix. A graph expansion is obtained from a given graph when an edge is replaced by a path. One of the key results is that the spectral radius strictly decreases under such an operation on an unweighted irreducible graph.The Perron-Frobenius theory does not directly apply in this situation since one is comparing spectral radii of adjacency matrices of different sizes. However, as demonstrated in [7] and [6], one can write equal size adjacency matrices for a graph and its expansion if one allows polynomial entries in the matrix. Polynomial matrices have also arisen as important tools for realization and classification problems in symbolic dynamics; see [1,3,8].More formally, let G = (V, E) be an irreducible directed graph with a finite vertex set V and a finite edge set E (we allow for multiple edges between two vertices). Let w : E → R + (for X ⊂ R, we use the notation X + to denote the set of elements of X which are strictly positive). We call the function w a weight function and the pair (G, w) a weighted graph. Let W = (w(i, j)) be the adjacency matrix for (G, w), i.e., w(i, j) is the sum of weights of edges from vertex i to vertex j. Now suppose that for each edge e ∈ E, we assign a positive integer α(e) and modify the graph by the following process: introduce α(e) − 1 new vertices to the graph, remove the edge e and replace e with a path P(e) of length α(e) which originates at the initial vertex of e, passes through the α(e) − 1 new vertices, and ends at the terminal vertex of e. If we assign the weight of w(e) to the first edge in P(e) and 1 to all others, we obtain a new weighted graph which we will denote (G α , w α ).
