549 research outputs found
The zero forcing polynomial of a graph
Zero forcing is an iterative graph coloring process, where given a set of
initially colored vertices, a colored vertex with a single uncolored neighbor
causes that neighbor to become colored. A zero forcing set is a set of
initially colored vertices which causes the entire graph to eventually become
colored. In this paper, we study the counting problem associated with zero
forcing. We introduce the zero forcing polynomial of a graph of order
as the polynomial , where is
the number of zero forcing sets of of size . We characterize the
extremal coefficients of , derive closed form expressions for
the zero forcing polynomials of several families of graphs, and explore various
structural properties of , including multiplicativity,
unimodality, and uniqueness.Comment: 23 page
A measure of centrality based on the spectrum of the Laplacian
We introduce a family of new centralities, the k-spectral centralities.
k-Spectral centrality is a measurement of importance with respect to the
deformation of the graph Laplacian associated with the graph. Due to this
connection, k-spectral centralities have various interpretations in terms of
spectrally determined information.
We explore this centrality in the context of several examples. While for
sparse unweighted networks 1-spectral centrality behaves similarly to other
standard centralities, for dense weighted networks they show different
properties. In summary, the k-spectral centralities provide a novel and useful
measurement of relevance (for single network elements as well as whole
subnetworks) distinct from other known measures.Comment: 12 pages, 6 figures, 2 table
Some new aspects of main eigenvalues of graphs
An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is non-orthogonal to the associated eigenspace. This paper explores some new aspects of the study of main eigenvalues of graphs, investigating specifically cones over strongly regular graphs and graphs for which the least eigenvalue is non-main. In this case, we characterize paths and trees with diameter-3 satisfying the property. We may note that the importance of
least eigenvalues of graphs for the equilibria of social and economic networks was recently uncovered in literature.publishe
Cutoff for non-backtracking random walks on sparse random graphs
A finite ergodic Markov chain is said to exhibit cutoff if its distance to
stationarity remains close to 1 over a certain number of iterations and then
abruptly drops to near 0 on a much shorter time scale. Discovered in the
context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now
believed to be rather typical among fast mixing Markov chains. Yet,
establishing it rigorously often requires a challengingly detailed
understanding of the underlying chain. Here we consider non-backtracking random
walks on random graphs with a given degree sequence. Under a general sparsity
condition, we establish the cutoff phenomenon, determine its precise window,
and prove that the (suitably rescaled) cutoff profile approaches a remarkably
simple, universal shape
On the maximum -spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices
For a connected graph , let be the adjacency matrix of and
be the diagonal matrix of the degrees of the vertices in . The
-matrix of is defined as \begin{align*} A_\alpha (G) = \alpha
D(G) + (1-\alpha) A(G) \quad \text{for any }. \end{align*}
The largest eigenvalue of is called the -spectral
radius of . In this article, we characterize the graphs with maximum
-spectral radius among the class of unicyclic and bicyclic graphs
of order with fixed girth . Also, we identify the unique graphs with
maximum -spectral radius among the class of unicyclic and bicyclic
graphs of order with pendant vertices.Comment: 16 page
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