Why do we consider a graph as distance-transitive graph when we talk about a perfect code in a graph? Indeed, When there is a non-trivial perfect code in a graph, is the graph a distance-transitive or distance-regular graph? Is there a graph which is not distance-regular but there is non-trivial perfect code in it?
2026-04-01 14:19:39.1775053179
Perfect code in graphs not distance-transitive
210 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 CODING-THEORY
- Solving overdetermined linear systems in GF(2)
- Inverting a generator matrix - Coding Theory
- Probability of a block error of the (N, K) Hamming code used for a binary symmetric channel.
- How to decode a Hadamard message that was encoded using the inner product method?
- How to decode a Hadamard message that was encoded using a generator matrix?
- Find the two missing digits in 10-ISBN code
- Characterize ideals in $\mathbb{F}_l[x]/(x-1) \oplus \mathbb{F}_l[x]/(\frac{x^p-1}{x-1})$
- Number of codes with max codeword length over an alphabet
- Dimension of ASCII code
- Prove how many errors CRC code can detect
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?
Classical coding theory studies perfect codes in Hamming graphs. Since Hamming graphs are distance transitive, it is reasonable to consider perfect codes in distance-regular graphs.
Next. some of the basic results, such as Lloyd's theorem extend very naturally to distance-regular graphs. This again provides evidence that it is reasonable to consider perfect codes in these graphs.
There are regular graphs that are not distance-regular but do have perfect 1-codes. If $A$ and $B$ are the adjacency matrices of a graph $X$ and its complement, the matrix $$ \begin{pmatrix}A&B\\ B&A\end{pmatrix} $$ is the adjacency matrix of a graph with a perfect 1-code of size two (in most cases). (There are many other examples, but these are the first that come to mind.)