Abstract. We investigate some of the properties and extensions of a dynamic innovation network model recently introduced in [37]. In the model, the set of efficient graphs ranges, depending on the cost for maintaining a link, from the complete graph to the (quasi-) star, varying within a well defined class of graphs. However, the interplay between dynamics on the nodes and topology of the network leads to equilibrium networks which are typically not efficient and are characterized, as observed in empirical studies of R&D networks, by sparseness, presence of clusters and heterogeneity of degree. In this paper, we analyze the relation between the growth rate of the knowledge stock of the agents from R&D collaborations and the properties of the adjacency matrix associated with the network of collaborations. By means of computer simulations we further investigate how the equilibrium network is affected by increasing the evaluation time τ over which agents evaluate whether to maintain a link or not. We show that only if τ is long enough, efficient networks can be obtained by the selfish link formation process of agents, otherwise the equilibrium network is inefficient. This work should assist in building a theoretical framework of R&D networks from which policies can be derived that aim at fostering efficient innovation networks. 1. Introduction. The

Let T(G) be the number of spanning trees in graph G. In this note we explore the asymptotics of T(G) for circulant graphs. The circulant graph Cs1,s2,···,sk n is the 2k regular graph with n vertices labelled 0,1,2, · · ·,n− 1, where node i has the 2k neighbors, (0 ≤ i ≤ n − 1) adjacent to vertices i + s1,i + s2, · · ·,i + sk mod n. In this note we give a closed formula for the asymptotic limit limn→ ∞ T(C s1,s2,···,sk n as a function of s1,s2,...,sn. We then extend this by permitting the si to be linear functions of n, i.e., we give a closed formula for lim n→ ∞ T C

We introduce a concept of similarity between vertices of directed graphs. Let GA and GB be two directed graphs with respectively nA and nB vertices. We define a nA nB similarity matrix S whose real entry s ij expresses how similar vertex i (in GA ) is to vertex j (in GB ) : we say that s ij is their similarity score. In the special case where GA = GB = G, the score s ij is the similarity score between the vertices i and j of G and the square similarity matrix S is the self-similarity matrix of the graph G. We point out that Kleinberg's "hub and authority" method to identify webpages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them. In analogy to Kleinberg, we show that our similarity scores are given by the components of a dominant vector of a non-negative matrix and we propose a simple iterative method to compute them. Potential applications of our similarity concept are manifold and we illustrate one application for the automatic extraction of synonyms in a monolingual dictionary.

number of spanning trees in a class of double fixed-step loop networks

A tree which has exactly one vertex of degree greater than two is said to be starlike. In spite of seemingly simple structure of these trees, not much is known about their spectral properties. In this paper, we introduce a generalization of the notion of cospectrality called m-cospectrality which turns out to be useful in constructing cospectral graphs. Based on this, we construct cospectral mates for some starlike trees. We also present a set of necessary and sufficient conditions for divisibility of the characteristic polynomial of a starlike tree by the characteristic polynomial of a path. AMS Subject Classification: 05C50.

We consider θ-graphs, that is, graphs obtained by subdividing the edges of the multigraph consisting of 3 parallel edges. It is shown that any θ-graph G is determined by the spectrum (the multiset of eigenvalues) except possibly when it contains a unique 4-cycle. AMS Subject Classification: 05C50.

A graph is said to be determined by the adjacency spectrum (DS for short) if there is no other nonisomorphic graph with the same spectrum. All connected graphs with index at most √ 2 + √ 5 are known. In this paper, we show that with few exceptions all of these graphs are DS. AMS Subject Classification: 05C50.

It is proved that the Cartesian product of an odd cycle with the complete graph on 2 vertices, is determined by the spectrum of the adjacency matrix. We also present some computational results on the spectral characterization of cubic graphs on at most 20 vertices. AMS Subject Classification: 05C50.

We investigate graphs whose signless Laplacian matrix has three distinct eigenvalues. We show that the largest signless Laplacian eigenvalue of a connected graph G with three distinct signless Laplacian eigenvalues is noninteger if and only if G = Kn −e for n ≥ 4, where Kn −e is the n vertex complete graph with an edge removed. Moreover, examples of such graphs are given. AMS Mathematics Subject Classification: 05C50.

We prove that for any integers k, m> 0 and any tree F with at least one edge, there exists a tree whose number of F-matchings is congruent to k modulo m as well as an analogous result for induced F-matchings. This answers a question of Alon, Haber and Krivelevich (The number of F-matchings in almost every tree is a zero residue, Electron. J. Combin. 18 (2011), #P30). 1