I have to prove that a certain tree has a certain structure, and I've gotten as far as proving that it has the correct Laplace spectrum. While trees are definitely not determined by the spectrum of their adjacency matrix, I haven't been able to find anything on whether they're determined by their Laplace spectrum, with multiplicity (and if not, which ones are.) Is this a settled question, and if so where can I find a resource on it?
2026-03-26 06:05:18.1774505118
Are trees determined by their Laplace spectrum?
169 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GRAPH-THEORY
- characterisation of $2$-connected graphs with no even cycles
- Explanation for the static degree sort algorithm of Deo et al.
- A certain partition of 28
- decomposing a graph in connected components
- Is it true that if a graph is bipartite iff it is class 1 (edge-coloring)?
- Fake induction, can't find flaw, every graph with zero edges is connected
- Triangle-free graph where every pair of nonadjacent vertices has exactly two common neighbors
- Inequality on degrees implies perfect matching
- Proving that no two teams in a tournament win same number of games
- Proving that we can divide a graph to two graphs which induced subgraph is connected on vertices of each one
Related Questions in SPECTRAL-GRAPH-THEORY
- Is a stable Metzler matrix minus a Metzler matrix with zero along diagonal also stable?
- Diagonally dominant matrix by rows and/or by columns
- Shape of the graph spectrum
- Let $G$ be a planar graph with $n$ vertices, then $\lambda_1(G) \leq −3 \lambda_n(G)$.
- How can one construct a directed expander graph with varying degree distributions (not d-regular)?
- book recommendation: differential equations on networks
- Do isomorphic graphs have same values for adjacency matrices and spectrum?
- Normalized Laplacian eigenvalues of a path graph
- Equitable partitions in the undirected graph
- Approximate discrete Laplacian with continuous Laplacian
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
No. See http://users.cecs.anu.edu.au/~bdm/papers/SpectralTrees.pdf for an infinite family of counterexamples. (Published in Ars Combinatoria 3 (1977) 219-232.) I should point out that this paper proves that the proportion of trees on $n$ vertices that are determined by the characteristic polynomials of the Laplacian matrix, the adjacency matrix, and a number of other matrices (all together), goes to zero as $n\to\infty$.