On the Eigenvalue Estimates for the Weighted Laplacian on Metric Graphs
نویسندگان
چکیده
It is shown that the eigenvalues of the equation −λ∆u = V u on a graph G of final total length |G|, with non-negative V ∈ L(G) and under appropriate boundary conditions, satisfy the inequality n2λn ≤ |G| ∫ G V dx, independently of geometry of a given graph. Applications and generalizations of this result are also discussed.
منابع مشابه
Eigenvalue Bracketing for Discrete and Metric Graphs
We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph La...
متن کاملOn Complementary Distance Signless Laplacian Spectral Radius and Energy of Graphs
Let $D$ be a diameter and $d_G(v_i, v_j)$ be the distance between the vertices $v_i$ and $v_j$ of a connected graph $G$. The complementary distance signless Laplacian matrix of a graph $G$ is $CDL^+(G)=[c_{ij}]$ in which $c_{ij}=1+D-d_G(v_i, v_j)$ if $ineq j$ and $c_{ii}=sum_{j=1}^{n}(1+D-d_G(v_i, v_j))$. The complementary transmission $CT_G(v)$ of a vertex $v$ is defined as $CT_G(v)=sum_{u in ...
متن کاملA sharp upper bound on the largest Laplacian eigenvalue of weighted graphs
We consider weighted graphs, where the edge weights are positive definite matrices. The Laplacian of the graph is defined in the usual way. We obtain an upper bound on the largest eigenvalue of the Laplacian and characterize graphs for which the bound is attained. The classical bound of Anderson and Morley, for the largest eigenvalue of the Laplacian of an unweighted graph follows as a special ...
متن کاملCounting the number of spanning trees of graphs
A spanning tree of graph G is a spanning subgraph of G that is a tree. In this paper, we focus our attention on (n,m) graphs, where m = n, n + 1, n + 2, n+3 and n + 4. We also determine some coefficients of the Laplacian characteristic polynomial of fullerene graphs.
متن کاملHeat kernel estimates on weighted graphs
We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates allow for a very explicit probabilistic interpretation and can be formulated in terms of a weighted metric. Interestingly, this metric is not equivalent to the intrinsic metric. The results Heat kernel estimates are a tool ...
متن کامل