We investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.

We explore the tunneling behavior of a quantum particle on a finite graph in the presence of an asymptotically large potential with two or three potential wells. The behavior of the particle is primarily governed by the local spectral symmetry of the graph around the wells. In the case of two wells the behavior is stable in the sense that it can be predicted from a sufficiently large neighborhood of the wells. However in the case of three wells we are able to exhibit examples where the tunneling behavior can be changed significantly by perturbing the graph arbitrarily far from the wells.

In a graph G, for a subset S of the vertex set, the induced subgraph determined by S has edge set consisting of all edges of G with both endpoints in S. The (edge) boundary, denoted by S consists of all edges containing one endpoint in S and one endpoint not in S.

This erratum will correct the classification of Theorem 1 in Lin-Lu-Yau, Comm. Anal. Geom., 2014, that misses the Triplex graph.

"In this paper we consider a variation of the classical Turán-type extremal problems. Let $S$ be an $n$ -term graphical sequence, and $\sigma(S)$ be the sum of the terms in $S$ . Let $H$ be a graph. The problem is to determine the smallest even $l$ such that any $n$ -term graphical sequence $S$ having $\sigma(S)\geq l$ has a realization containing $H$ as a subgraph. Denote this value $l$ by $\sigma(H,n)$ . We show $\sigma(C_{2m+1},n)=m(2n-m-1)+2$ , for $m\geq 3$ , $n\geq 3m$ ; $\sigma(C_{2m+2},n)=m(2n-m-1)+4$ , for $m\geq 3$ , $n\geq 5m-2$ .''