On spectral radii of unraveled balls

Given a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We prove a lower bound on the maximum spectral radius of unraveled balls of fixed radius, and we show, among other things, that if the average degree of $G$ after deleting any ball of radius $r$ is at least $d$ then its second largest eigenvalue is at least $2\sqrt{d-1}\cos(\frac{\pi}{r+1})$.Comment: 8 pages, accepted to J. Comb. Theory B, corrections suggested by the referees have been incorporate

On the α\alpha-spectral radius of graphs

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix. The $\alpha$-spectral radius of $G$ is the largest eigenvalue of $A_{\alpha}(G)$. We give upper bounds for $\alpha$-spectral radius for unicyclic graphs $G$ with maximum degree $\Delta\ge 2$, connected irregular graphs with given maximum degree and and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with second maximum $\alpha$-spectral radius among trees, and the unique tree with maximum $\alpha$-spectral radius among trees with given diameter. For a graph with two pendant paths at a vertex or at two adjacent vertex, we prove results concerning the behavior of the $\alpha$-spectral radius under relocation of a pendant edge in a pendant path. We also determine the unique graphs such that the difference between the maximum degree and the $\alpha$-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature

The spectral radius and the maximum degree of irregular graphs

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of adding or removing few edges on the spectral radius of a regular graph.Comment: 10 pages, 1 figure, submitted to EJC on January 20, 200

Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues

It was recently shown by Lubetzky and Peres (2016) and by Sardari (2018) that Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the simple random walk in optimal time and have optimal almost-diameter. We prove that this spectral condition can be replaced by a weaker condition, the Sarnak-Xue density of eigenvalues property, to deduce similar results. We show that a family of Schreier graphs of the $SL_{2}\left(\mathbb{F}_{t}\right)$-action on the projective line satisfies the Sarnak-Xue density condition, and hence exhibit the desired properties. To the best of our knowledge, this is the first known example of optimal cutoff and almost-diameter on an explicit family of graphs that are neither random nor Ramanujan

Random spherical graphs

This work addresses a modification of the random geometric graph (RGG) model by considering a set of points uniformly and independently distributed on the surface of a $(d-1)$-sphere with radius $r$ in a $d-$dimensional Euclidean space, instead of on an unit hypercube $[0,1]^d$ . Then, two vertices are connected by a link if their great circle distance is at most $s$. In the case of $d=3$, the topological properties of the random spherical graphs (RSGs) generated by this model are studied as a function of $s$. We obtain analytical expressions for the average degree, degree distribution, connectivity, average path length, diameter and clustering coefficient for RSGs. By setting $r=\sqrt{\pi}/(2\pi)$, we also show the differences between the topological properties of RSGs and those of two-dimensional RGGs and random rectangular graphs (RRGs). Surprisingly, in terms of the average clustering coefficient, RSGs look more similar to the analytical estimation for RGGs than RGGs themselves, when their boundary effects are considered. We support all our findings by computer simulations that corroborate the accuracy of the theoretical models proposed for RSGs.Comment: 28 pages, 14 figure

Hitting and commute times in large graphs are often misleading

Next to the shortest path distance, the second most popular distance function between vertices in a graph is the commute distance (resistance distance). For two vertices u and v, the hitting time H_{uv} is the expected time it takes a random walk to travel from u to v. The commute time is its symmetrized version C_{uv} = H_{uv} + H_{vu}. In our paper we study the behavior of hitting times and commute distances when the number n of vertices in the graph is very large. We prove that as n converges to infinty, hitting times and commute distances converge to expressions that do not take into account the global structure of the graph at all. Namely, the hitting time H_{uv} converges to 1/d_v and the commute time to 1/d_u + 1/d_v where d_u and d_v denote the degrees of vertices u and v. In these cases, the hitting and commute times are misleading in the sense that they do not provide information about the structure of the graph. We focus on two major classes of random graphs: random geometric graphs (k-nearest neighbor graphs, epsilon-graphs, Gaussian similarity graphs) and random graphs with given expected degrees (in particular, Erdos-Renyi graphs with and without planted partitions

Spectral Gap of Random Hyperbolic Graphs and Related Parameters

Random hyperbolic graphs have been suggested as a promising model of social networks. A few of their fundamental parameters have been studied. However, none of them concerns their spectra. We consider the random hyperbolic graph model as formalized by [GPP12] and essentially determine the spectral gap of their normalized Laplacian. Specifically, we establish that with high probability the second smallest eigenvalue of the normalized Laplacian of the giant component of and $n$-vertex random hyperbolic graph is $\Omega(n^{-(2\alpha-1)}/D)$, where $\frac12<\alpha<1$ is a model parameter and $D$ is the network diameter (which is known to be at most polylogarithmic in $n$). We also show a matching (up to a polylogarithmic factor) upper bound of $n^{-(2\alpha-1)}(\log n)^{1+o(1)}$. As a byproduct we conclude that the conductance upper bound on the eigenvalue gap obtained via Cheeger's inequality is essentially tight. We also provide a more detailed picture of the collection of vertices on which the bound on the conductance is attained, in particular showing that for all subsets whose volume is $O(n^{1-\varepsilon})$ the obtained conductance is with high probability $\Omega(n^{-(2\alpha-1)\varepsilon+o(1)})$. Finally, we also show consequences of our result for the minimum and maximum bisection of the giant component.Comment: 44 pages, 3 figure

The hydrogen identity for Laplacians

For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, where H=(d+d^*)^2 is the sign-less Hodge Laplacian defined by sign-less incidence matrix d and where L is the connection Laplacian. Any spectral information about L directly leads to estimates for the Hodge Laplacian H=(d+d^*)^2 and allows to estimate the spectrum of the Kirchhoff Laplacian H_0=d^* d. The hydrogen identity implies that the random walk u(n) = L^n u with integer n solves the one-dimensional Jacobi equation Delta u=H^2 with (Delta u)(n)=u(n+2)-2 u(n)+u(n-2). Every solution is represented by such a reversible path integral. Over a finite field, we get a reversible cellular automaton. By taking products of complexes such processes can be defined over any lattice Z^r. Since L^2 and L^(-2) are isospectral, by a theorem of Kirby, the matrix L^2 is always similar to a symplectic matrix if the graph has an even number of simplices. The hydrogen relation is robust: any Schr\"odinger operator K close to H with the same support can still can be written as $K=L-L^{-1}$ where both L(x,y) and L^-1(x,y) are zero if x and y do not intersect.Comment: 29 pages, 8 figure