Old and new results on algebraic connectivity of graphs

On conjectures involving second largest signless Laplacian eigenvalue of graphs

The spectra of the adjacency matrix and Laplacian matrix for some balanced trees

https://digitalcommons.sacredheart.edu/cgi/viewcontent.cgi?article=1045&context=math_fac

On graphs with equal algebraic and vertex connectivity

We consider the change-point detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two connected induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) statistics and relate it to the problem of finding a sparsest cut in a graph. We develop a tractable relaxation of the GLR statistic based on the combinatorial Laplacian of the graph, which we call the spectral scan statistic, and analyze its properties. We show how its performance as a testing procedure depends directly on the spectrum of the graph, and use this result to explicitly derive its asymptotic properties on few significant graph topologies. Finally, we demonstrate both theoretically and by simulations that the spectral scan statistic can outperform naive testing procedures based on edge thresholding and $\chi^2$ testing.

Changepoint Detection over Graphs with the Spectral Scan Statistic

In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set Si,n to be the set of all integers from 0 to n, excluding i. If there exists a graph whose Laplacian matrix has this set as its eigenvalues, we say that this set is Laplacian realizable. We investigate the sets Si,n that are Laplacian realizable, and the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues. We characterize those i < n such that Si,n is Laplacian realizable, and show that for certain values of i, the set Si,n is realized by a unique graph. Finally, we conjecture that Sn,n is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n. © 2005 Wiley Periodicals, Inc. J Graph Theory

On graphs whose Laplacian matrices have distinct integer eigenvalues

In this paper, quite tight lower and upper bounds are obtained on the algebraic connectivity, namely, the second-smallest eigenvalue of the Laplacian matrix, of an unweighted balanced binary tree with k levels and hence n = 2 1 vertices This is accomplished by considering the inverse of a matrix of order k 1 readily obtained from the Laplacian matrix It is shown that the algebraic connectivity is 1=(2 2k + 3) +O(1=2)

/pdf/tight-bounds-on-the-algebraic-connectivity-of-a-balanced-45jak8uv19.pdf

Jason J. Molitierno

Papers

On graphs with equal algebraic and vertex connectivity

On graphs whose Laplacian matrices have distinct integer eigenvalues

Tight bounds on the algebraic connectivity of a balanced binary tree

On the algebraic connectivity of graphs as a function of genus

On trees with perfect matchings