Edge connectivity and the spectral gap of combinatorial and quantum graphs

Abstract

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds is the well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Lévy. All proofs are general enough to yield corresponding estimates for the p-Laplacian and allow us to identify the minimizers.Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are "asymptotically correct", i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.

Surgery principles for the spectral analysis of quantum graphs

“…Remark 3.19. Part (3) has already appeared in the literature in a less general form, most recently in [BKKM17]. The other statements are, to the best of our knowledge, completely new.…”

“…For a general compact metric graph of total length L, this eigenvalue was shown by Nicaise [Nic87] to be no smaller than π 2 /L 2 , with equality if and only if G is a path (i.e., interval); see [Sol02,Fri05a,KN14] for further proofs. Recently, Band and Lévy [BaLe17] obtained a stronger lower bound under the assumption that the graph is doubly connected: the non-trivial eigenvalue is no lower than 4π 2 /L 2 ; see also [BKKM17] for a sharper estimate in the case of higher connectivities.…”

Limits of quantum graph operators with shrinking edges

“…Continuous dependence of eigenvalues on edge lengths is a fundamental issue in the spectral theory of quantum graphs [BK,M14]. In particular, it is vital to spectral shape optimization problems which have received much attention recently (see for example [F05,EJ,KKM,BRV,KKMM,DR,BL,BKKD,R17,Ar] and references therein). In such optimization problems achieving extremum often requires redistribution of volume (edge length) from one edge to another.…”

Surgery of Graphs: M-Function and Spectral Gap

“…An alternative approach was developed in [5,8,9,11], where spectral properties of graphs in relation to their connectivity are studied.…”

“…Therefore it appears important to study behaviour of the spectrum under the change of the metric graph. Different transformations of the underlying metric graph have been considered [3][4][5][6][7][8][9][10][11]. Our goal today is to understand what happens to the spectrum if the graph is cut into two or more pieces or if two or more graphs are glued into one.…”