All Questions
Tagged with adjacency-matrices spectral-graph-theory
9 questions
24
votes
6
answers
2k
views
Factorization of the characteristic polynomial of the adjacency matrix of a graph
Let $G$ be a regular graph of valence $d$ with finitely many vertices, let $A_G$ be its adjacency matrix, and let $$P_G(X)=\det(X-A_G)\in\mathbb{Z}[X]$$ be the adjacency polynomial of $G$, i.e., the ...
- 47.4k
11
votes
1
answer
2k
views
Eigenvalues of the complement of a graph
Let $A$ and $\widetilde A$ be the adjacency matrices of a graph $G$ and of its complement, respectively.
Is there any relation between the eigenvalues of $A + \widetilde A$ and the eigenvalues of $A$ ...
- 1,269
8
votes
3
answers
8k
views
Spectrum of an adjacency matrix
The adjacency matrix of a non-oriented connected graph is symmetric, hence its spectrum is real.
If the graph is bipartite, then the spectrum of its adjacency matrix is symmetric about 0. A few ...
- 3,388
6
votes
1
answer
517
views
Non-diagonalizability of the adjacency matrix of a directed graph
Let $G$ be a directed graph with no multiple edges or loops and let $P_i$ be its vertices. Let $A$ be the corresponding adjacency matrix of $G$, i.e. $a_{i,j}=1$ if and only if there is a directed ...
- 4,058
5
votes
0
answers
222
views
(How) does the spectral gap of the $n\times n$ Rubik's cube close with $n$?
Consider the spectrum of the adjacency matrix $A$ of the Cayley graph of the standard, 3x3x3 Rubik's cube generated with the usual quarter-turn and half-turn twists of each face (the Singmaster ...
- 2,185
4
votes
1
answer
207
views
Digraphs with unique walk of length $k$ between any two vertices
Let $G$ be a digraph such that there is an unique directed walk of length $k$ between any two vertices.
Equivalently, if $A$ is the adjacency matrix of $G$, then $A^k$ is the matrix with all entries $...
- 3,446
2
votes
1
answer
316
views
When does a row standardized adjacency matrix have a real spectrum?
A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions ...
- 6,001
2
votes
0
answers
55
views
Regularize a graph while embedding the spectrum of adjacency matrix
Given an irregular graph $G$ whose maximum degree is $d$, I am interested in producing a new graph $G'$ which is regular and has the spectrum of the adjacency spectrum of $G$ embedded in the spectrum ...
- 21
2
votes
0
answers
159
views
Formulas to determine the value of graph energy with addition or deletion of edges
If $G$ is a graph, then the graph energy of $G$ denoted by $E(G)$ is defined as the sum of absolute values of eigenvalues of the adjacency matrix of $G$. It is known that $E(G)\geq E(G-v)$, where $ ...
- 203