منابع مشابه
3 Planarity via Generalized Laplacians
The main topic of this course has been the unexpected connection between graph theory and spectral theory: we’ve seen the relationships between graph spectra and connectivity, bipartiteness, independent sets, colorings, trees, cuts, flows, and others. Today we see another connection with a standard concept in graph theory: planarity. In many branches of mathematics, it is frequently helpful to ...
متن کاملHeat Kernel Asymptotics for Roots of Generalized Laplacians
We describe the heat kernel asymptotics for roots of a Laplace type operator ∆ on a closed manifold. A previously known relation between the Wodzicki residue of ∆ and heat trace asymptotics is shown to hold pointwise for the corresponding densities.
متن کاملRigidity of generalized laplacians and some geometric applications
Every generalized laplacian L defined on a manifold M determines a sheaf of "L-harmonic" sections namely the sheaf of local solutions of Lu = 0. We study the converse problem: to what extent this sheaf determines the operator. Our main result states that the sheaf of L-harmonic sections determines the operator up to a conformal factor. Moreover, when the operator is a covariant laplacian and th...
متن کاملGeneralized Laplacians and First Transit Times for Directed Graphs
In this paper, we extend previous results on average commute-times for undirected graphs to fully-connected directed graphs, corresponding to irreducible Markov chains. We introduce an unsymmetrized generalized Laplacian matrix and show how its pseudo-inverse directly yields the one-way first-transit times and round-trip commute times with formulas almost matching those for the undirected graph...
متن کاملDirected Hamiltonicity and Out-Branchings via Generalized Laplacians
We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant 0 < λ < 1 and prime p we can count the Hamiltonian cycles modulo pb(1−λ) n 3p c in expected time less than...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1956
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1956-0078472-8