I would like to know the graphs for which the (node-node) adjacency matrix is totally unimodular. Is the following true?
The adjacency matrix of G is totally unimodular if and only if G is a tree.
2026-03-25 08:09:21.1774426161
When is the adjacency matrix of a simple undirected graph totally unimodular?
736 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 TREES
- Explanation for the static degree sort algorithm of Deo et al.
- Finding height of a $k$-ary tree
- Clique-width of a tree
- count "informative" paths in tree
- If the weight of edge E $e$ of an MST is decreased by $\delta$. Could total weight of MST decrease by more than $\delta$.
- Probability of two randomly selected leaves of a tree to be connected only at the root
- Proof in graph theory: maximum degree and number of leaves.
- Graph Theory: Number of vertices in a tree.
- The number of, and an enumeration for, the set of full subtrees of the full complete binary tree
- Is the maximum link length function convex?
Related Questions in ADJACENCY-MATRIX
- Shape of the graph spectrum
- Use the definition of matrix multiplication to explain why the analogous result holds for any entry of Adjacency Matrix $A^2$.
- Do isomorphic graphs have same values for adjacency matrices and spectrum?
- Edge-to-edge incidence structure of a graph
- Is there an approximation to a matrix $V = (I-cA)^{-1}$ where $I$ is the identity matrix and $A$ is an adjacency matrix of a connected graph?
- Is it possible to normalize a symmetric matrix without breaking symmetry?
- Spectral radius of a complete bipartite graph
- What is a suitable index to express similarity in two observations of the same set of variables containing ratios?
- How to find Eulerian path in the given graph?
- Spectral radius of a complete tripartite graph
Related Questions in UNIMODULAR-MATRICES
- Does this property imply total unimodularity?
- Is this block matrix totally unimodular?
- If matrix $A$ is totally unimodular, then matrix $\begin{bmatrix} A &\pm A\end{bmatrix}$ is also totally unimodular
- Find explicitely an element of SL$_2(\mathbb Z)$
- (reference-request) Theorem of Frobenius regarding transforming integer-valued matrix in diagonal matrix with divisibility conditions
- Proof $D_2^{-1}D_1$ is a unimodular matrix.
- Ring of invariants of unimodular matrices acting on real square matrices
- Reference for a result in Abelian Group Theory
- Totally Unimodular Matrices
- How can I compute a 3 by 3 unimodular matrix which produces an infinite number of Fermat near misses?
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?
EDIT: See Akbari and Kirkland, On unimodular graphs, Linear Algebra and its Applications, Volume 421, Issue 1, 1 February 2007, Pages 3-15. According to the abstract, "Graphs whose adjacency matrices are totally unimodular are also characterized."
That paper is available via https://www.sciencedirect.com/science/article/pii/S0024379506004708
Ignore what follows, left here for historical reasons:
Wikipedia says, "The unoriented incidence matrix of a bipartite graph, which is the coefficient matrix for bipartite matching, is totally unimodular (TU). (The unoriented incidence matrix of a non-bipartite graph is not TU.)"