Spectral Radius Of Digraphs Research Articles

Bounds on the Spectral Radius of Digraphs from Subgraph Counts

The spectral radius of a directed graph is a metric that can only be computed when the structure of the network is completely known. However, in many practical scenarios, it is not possible to exactly retrieve the whole structure of the network; hence, the exact value of the spectral radius is not computable. Even in these scenarios, it is typically possible to extract local structural properties of a network using, for example, graph crawlers. In this paper, we develop a novel measure-theoretic framework to upper and lower bound the spectral radius of a directed graph using local structural information, in particular, using the counts of a collection of small subgraphs or motifs. Our framework is based on recent results relating the multivariate moment problem with semidefinite programming. Using these results, we develop a hierarchy of (small) semidefinite programs whose solutions provide upper and lower bounds on the spectral radius of a directed graph using, solely, subgraph and motif counts. We numerically validate the quality of our bounds using both random and real-world directed graphs.

On the distance spectral radius of digraphs with given diameter

The distance spectral radius of a strongly connected digraph G is the eigenvalue of its distance matrix with the largest modulus. Let denote the set of strongly connected digraphs with order n and diameter d. In this paper, we completely determine the strongly connected digraphs minimizing among all strongly connected digraphs with order n and diameter d, for d = 1, 2, 3, 4, 5, 6, 7, n−1. We also propose a conjecture about the minimum distance spectral radius among all strongly connected digraphs with given diameter .

Imprimitivity index of the adjacency matrix of digraphs

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ(G)≥3, then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D1 is a primitive regular orientation and D2 is a non-regular orientation of a given graph, then for sufficiently large t, the number of closed walks of length t in D1 is more than the number of closed walks of length t in D2.

Colorings and spectral radius of digraphs

We determine the digraphs that have the minimum and second minimum spectral radius among all strongly connected digraphs with given order and dichromatic number. This solves a problem posed by Lin, Shu, Wu, and Yu.

Spectral radius of digraphs with given dichromatic number

Let D be a digraph with vertex set V ( D ) . A partition of V ( D ) into k acyclic sets is called a k -coloring of D . The minimum integer k for which there exists a k -coloring of D is the dichromatic number χ ( D ) of the digraph D . Denote G n , k the set of the digraphs of order n with the dichromatic number k ≥ 2 . In this note, we characterize the digraph which has the maximal spectral radius in G n , k . Our result generalizes the result of [8] by Feng et al.

Computing the Hausdorff dimension of subshifts using matrices

On a subshift of finite type (SFT) we introduce a pseudometric d given by a nonnegative matrix B satisfying the cycle condition. We show that the Hausdorff dimension of this SFT with respect to d is given by the Mauldin-Williams formula. If the ratio of the logarithms of any two nonzero entries of B is rational, we show that this Hausdorff dimension can be expressed essentially in terms of the logarithm of the spectral radius of a certain digraph. We apply our results to the Hausdorff dimension of the limit set of finitely generated free groups of isometrics of infinite trees. To each finitely generated subgroup G of a given finitely generated free group F , we attach an invariant p ( G ), which gives the rate of growth of all words G of length l at most with respect to a fixed set of minimal generators of F . We show that p ( G ) is the spectral radius of a digraph Δ( G ) induced by G . Then H ⩽ G ⩽ F ⇏ p ( G ) ⩾ p ( H ). Moreover, p ( G ) = p ( H ) ⇔ [ G : H ] < ∞.

Computing the Hausdorff dimension of subshifts using matrices

On a subshift of finite type (SFT) we introduce a pseudometric d given by a nonnegative matrix B satisfying the cycle condition. We show that the Hausdorff dimension of this SFT with respect to d is given by the Mauldin-Williams formula. If the ratio of the logarithms of any two nonzero entries of B is rational, we show that this Hausdorff dimension can be expressed essentially in terms of the logarithm of the spectral radius of a certain digraph. We apply our results to the Hausdorff dimension of the limit set of finitely generated free groups of isometrics of infinite trees. To each finitely generated subgroup G of a given finitely generated free group F, we attach an invariant p( G), which gives the rate of growth of all words G of length l at most with respect to a fixed set of minimal generators of F. We show that p( G) is the spectral radius of a digraph Δ( G) induced by G. Then H ⩽ G ⩽ F ⇏ p( G) ⩾ p( H). Moreover, p( G) = p( H) ⇔ [ G: H] < ∞.