Questions tagged [algebraic-graph-theory]

Algebraic methods in Graph Theory; the linear algebra method, graph homomorphisms, group theoretic methods (for example Cayley graphs), and graph invariants. For graph eigenvalue problems use the spectral-graph-theory tag. For strongly regular graphs use the strongly-regular-graph tag. For Kneser graphs use the kneser-graph tag.

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Homomorphisms relationship with Graph Degeneracy

Let $H, G$ be finite undirected graphs. We say that $H$ is $r$-degenerate if there exists an ordering of the vertices of $H$ such that the back degree of every vertex is at most $r$. This is ...
11 votes
5 answers
1k views

Which directed graphs have a normal adjacency matrix?

I am working on a problem in matrix analysis and I am looking for certain types of normal matrices. I suspect that these "special" normal matrices arise as adjacency matrices of certain graphs. My ...
3 votes
0 answers
210 views

Efficient way to calculate Smith Normal Form of large integer matrices

I am interested in calculating the Smith Normal Form for Laplacian matrices of hypercube graphs. Using the elementary divisors method from SAGE, I was able calculate up to the 11-cube (which has a $2^{...
0 votes
0 answers
122 views

On the absolute difference of the Laplacian eigenvalues of an unbalanced signed graph and its underlying graph

Let $\Sigma=(G,\sigma)$ be an unbalanced signed graph with the underlying connected graph $G=(V,E)$ and $\sigma:E\rightarrow \{-1,1\}$, the signing function. Let the Laplacian eigenvalues of $\Sigma$ ...
11 votes
1 answer
433 views

Correspondence between matrix multiplication and a graph operation of Lovász

In his book "Large networks and graph limits", Lovász describes a multiplication operation (he calls it concatenation) on "bi-labeled graphs". An $(m,n)$ bi-labeled graph is a ...
9 votes
3 answers
560 views

Generating (or availability of) large strongly regular graphs

Are there collections of already generated large strongly regular graphs available to download? By large I mean $n \geq 200$ where $n$ is the number of vertices. I found Ted Spence's page on srgs, ...
8 votes
0 answers
149 views

Partial order on graphs induced by homomorphism counts

For graphs $F$ and $G$, let $\hom(F,G)$ denote the number of homomorphisms (adjacency preserving maps) from $F$ to $G$. Define a relation $\le_{\hom}$ on (isomorphism classes of) graphs as $G \le_{\...
17 votes
1 answer
1k views

Bicycles and spanning trees of graphs

A spanning tree in a graph is a connected spanning subgraph with no cycles; it is well known that the number of spanning trees can be found by taking the determinant of a certain matrix related to the ...
6 votes
1 answer
437 views

An algebraic view of graph reconstruction

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\Coh{Coh}$I have broken my question into a few sections for clarity and to provide sufficient context to the problem. I apologize for the length. The ...
1 vote
1 answer
225 views

Do the cycles containing a fixed edge generate the cycle space of a graph?

Let $G$ be a $2$-connected graph and for $e \in E(G)$ denote by $\mathcal{C_e}$ the set of all cycles of $G$ containing the edge $e$. For what set of edges does $\mathcal{C_e}$ contain a basis of the ...
15 votes
1 answer
506 views

Reference request: Moore graphs

It is clear that the term Moore graph was coined by Hoffman and Singleton in their paper On Moore graphs with diameters $2$ and $3$, where they write E. F. Moore has posed the problem of describing ...
5 votes
2 answers
292 views

Characterizing graphs whose incidence matrix has the all ones vector in its row span

Suppose we have a simple connected graph $G=(V,E)$. Then let $A$ be its $|E|\times |V|$ incidence matrix. Here I am considering the unoriented incidence matrix. I want to know when the row span of $A$ ...
3 votes
1 answer
171 views

When is a $k$-distance-transitive graph already distance-transitive?

Call a (finite and connected) graph $k$-distance-transitive if its symmetry group acts transitively on the pairs in each one of the sets $$D_\delta:=\{(i,j)\in V\times V\mid \mathrm d(i,j)=\delta\},\...
8 votes
3 answers
938 views

Classes of graphs for which isospectrum implies isomorphism?

The spectrum of a graph is the (multi)set of eigenvalues of its adjacency matrix (or Laplacian, depending on what you're interested in). In general, two non-isomorphic graphs might have the same ...
1 vote
0 answers
120 views

halved and folded hypercube duality

Notation. Consider the group $\Gamma=\mathbb{Z}_2^n$. I will denote the group operation aditively and by $\epsilon_i=(0,\dots,0,1,0,\dots,0)$ I denote the canonical generators. Let's define also $\...
1 vote
0 answers
100 views

Varieties determined by a characteristic-type of polynomial with the structure of an underlying graph

While writing my master thesis, following problem came up: Given a digraph $G$ with edges $e_1,..,e_n$ and a given a $n\times n$- matrix $A\in\mathbb{C}^{n\times n}$ such that $A_{ij}=0$ if the ending ...
9 votes
2 answers
471 views

Moore graphs and finite projective geometry

In a comment on a blog post from 2009 about the hypothetical Moore graph(s) of degree 57 and girth 5, Gordon Royle offered the following observation (reproduced here in full for the sake of ...
2 votes
0 answers
182 views

The rank of a Laplacian-type matrix

Suppose that $M$ is an integer, symmetric matrix of order $n>2$ with the positive integers $K_1,\dotsc,K_n$ on its main diagonal, and with all the off-diagonal elements equal to $0$ or $1$ so that ...
3 votes
0 answers
70 views

how do I find eigenvalues of Cayley graph for one subset given a different subset

How do I find eigenvalues for the adjacency matrix of Cayley graph $X(S_n,S)$ where $S_n$ is the symmetric group of order $n$ and $S$ is the set of transpositions $(i,i+1)$, if the eigenvalues of the ...
1 vote
0 answers
70 views

Algebraic Formulation of Graph Reconstruction [closed]

Background Let $G$ be a finite graph on $v$ vertices. The deck, $D(G)$, of $G$ is the multi-set of vertex-deleted subgraphs of $G$. The Graph Reconstruction Conjecture asserts that for finite graphs $...
1 vote
1 answer
158 views

Lower bound for $\vert \det A \vert $ for the adjacency matrix of regular graphs

Assume $G$ is a simple $k$-regular graph of order $n$ with adjacency matrix $A$ which is non-singular. Does anyone know some lower bounds for $\vert \det (A) \vert$ with respect to $n$, $k$ or both? ...
2 votes
1 answer
118 views

A diameter 2 arc-transitive graph whose complement is not arc-transitive?

A graph $G=(V,E)$ is arc-transitive if its symmetry group acts transitively on ordered pairs of adjacent vertices. In general, the complement of an arc-transitive graph is not arc-transitive. But I ...
1 vote
1 answer
87 views

Total behaviour of graph spectrum

Let $\mathcal{G}$ be the set of all finite connected simple graphs minus the complete graphs. For any $G\in \mathcal{G}$, let $\lambda_{\geq0}(G)$ denotes the smallest positive adjacency eigenvalue of ...
3 votes
0 answers
121 views

Vertex in a graph whose stabilizer (in a given group $\Gamma$ of automorphisms) does not fix any neighbour vertex?

I know next to nothing about graph theory, but I did recently use the concept of graphs and groups acting on them to formalize the proof of a statement that has a priori nothing to do with graphs. I ...
1 vote
0 answers
142 views

Symmetric subgraph configurations

Let $G,H$ be two simple graphs. Let's call a subgraph of $H$ that is isomorphic to $G$ a $G$-subgraph. Consider the following construction: Construction: Let $\mathcal G=\mathcal G(G,H)$ be a graph ...
0 votes
1 answer
144 views

Elusive groups and vertex-transitive graphs

This question is pertaining to finite connected vertex-transitive graphs. I recently read "Transitive permutation groups without semiregular subgroup" by Cameron, Giudici, Jones, Kantor, Klin, ...
5 votes
1 answer
148 views

For what graph does the following algebraic property hold?

Let $G=(V,E)$ be a simple graph. My question: For what graph $G$, does there exist a permutation $\sigma$ on $V$ such that $$\prod_{uv\in E}(x_{\sigma(u)}-x_{\sigma(v)})=-\prod_{uv\in E}(x_u-x_v)?$$ ...
2 votes
1 answer
497 views

History of algebraic graph theory

I need a source about the history of algebraic graph theory. I mean for solving which problems or responding to what needs it was created? Indeed, I want to write a note about the history of the ...
8 votes
2 answers
582 views

Does the clique-coclique bound hold for all walk-regular graphs?

The clique-coclique bound is said to hold for a simple graph $G$ on $n$ vertices if $\lvert \omega(G) \rvert \lvert \alpha(G) \lvert \leq n$, letting $\omega(G)$ and $\alpha(G)$ denote its clique and ...
4 votes
1 answer
291 views

Are there graphs with irrational eigenvalues which are all $>1$?

The eigenvalues associated to a graph's adjacency matrix are necessarily algebraic integers, because the adjacency matrix itself is entirely integer. I'm curious as to whether it's possible to have ...
0 votes
1 answer
124 views

Chromatic Polynomials of Circulant Graph With Two Parameters

I have been working with the chromatic polynomials of circulant graphs of prime order $p$ with two distinct parameters, i.e. $P_{p,i,j}(x):= P(C_{p}(i,j),x)$ with $1 \leq i \neq j \leq \ n/2.$ In ...
2 votes
1 answer
274 views

Bounds on singular values of invertible 0-1 matrices

I'm interested in considering digraphs from an algebraic perspective, which leads me to the following question. Consider an invertible 0-1 matrix of shape $n \times n$. What lower and upper bounds ...
1 vote
0 answers
44 views

The number of Laplacian eigenvalues of a graph in interval [k,n]

There are several upper and lower bounds for $m_G[2,n]$ (the number of Laplacian eigenvalues of a graph $G$ with $n$ vertices in the interval $[2,n]$). I want to know whether there exists any bound ...
6 votes
2 answers
534 views

Do perfect matching(s) have signatures in the graph eigenvalues?

If the edges of a bipartite graph are such that they can be seen as a disjoint union of perfect matchings then will this somehow reflect in the eigenvalues of the Laplacian? It would be helpful to ...
1 vote
1 answer
117 views

Quantified imbalance in signed graphs

Let $G=(V,E)$ be an $n$-vertex simple undirected graph. A signing of the graph is a function $s:E \to \{1,-1\}$, and $(G,s)$ is a signed graph. That is, we label each edge of the graph with $1$ or $-1$...
5 votes
1 answer
144 views

Inertia of a class of Cayley graphs

Let $H^n_2(d)$ be the Cayley graph with vertex set $\{0,1\}^n$ where two strings form an edge iff they have Hamming distance at least $d$. What is the inertia of these graphs, that is, the numbers of ...
9 votes
1 answer
472 views

Relationship of Weisfeiler-Lehman algorithm to weak isomorphism of coherent algebras

A coherent algebra is a matrix algebra (over $\mathbb{C}$) closed under conjugate transpose and Schur (entrywise) product, and that contains the identity matrix $I$ and all ones matrix $J$. Given ...
4 votes
1 answer
563 views

Smallest pair of non-isomorphic graphs equivalent under the Weisfeiler-Leman algorithm

The (2-dimensional) Weisfeiler-Leman algorithm is a method for partitioning the ordered pairs of vertices of a graph in a canonical way which gives rise to a powerful graph invariant (see for instance ...
4 votes
0 answers
296 views

For what (other) families of graphs does the clique-coclique bound hold?

For a graph $G$, let $\omega(G)$ and $\alpha(G)$ denoted the clique and independence numbers of $G$ respectively. For some families of graphs, e.g. vertex transitive graphs, it is known that $\alpha(G)...
3 votes
1 answer
68 views

Are cospectral signed graphs with identical underlying graph necessarily switching-equivalent?

I'm working with signed graphs and I don't know the answer to the following question. Also, I couldn't find the answer anywhere. Question: If we have two signed graphs with the same underlying graph ...
1 vote
1 answer
314 views

Determinant of incidence matrix of a unicyclic unbalanced signed graph

While reading a paper on unicyclic unbalanced signed graphs, I met the following fact: The determinant of the incidence matrix of a unicyclic unbalanced graph (i.e. the cycle of the graph has an ...
7 votes
2 answers
413 views

Automorphism group of a special commuting graph

Suppose $S_6$ is the symmetric group on six letters and let $X$ denote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ with vertex set $X$ and edges precisely the 2-element subsets ...
1 vote
0 answers
124 views

graphs with semiregular automorphisms

I need some "well-known" non-regular finite graphs (at least two vertices have different valency) whose automorphism groups contain a non-trivial subgroup that acts on the vertices semi-regularly (i....
1 vote
1 answer
276 views

Automorphism group of a graph

Suppose $\Gamma$ is a simple graph and $G=\mathrm{Aut}(\Gamma)$ is the automorphism group of $\Gamma$. If $G$ stabilizes a subgraph $\Gamma_1$,, and $G_0$ is the point-wise stabiliser of the set $V(\...
3 votes
1 answer
251 views

If L is the laplacian matrix of an undirected graph, and D is a diagonal matrix, what does the cofactor of L+D look like?

We know (e.g. [Godsil, Royle: Algebraic Graph Theory, Lemma 13.2.3]) that any cofactor of the Laplacian matrix of a graph is constant, and is equal to the number of spanning trees of the graph. How do ...
3 votes
1 answer
187 views

Imbalance in a Signed Graph

Let $G=(V,E)$ be an $n$-vertex simple undirected graph. A signing of the graph is a function $s:E \to \{1,-1\}$, and $(G,s)$ is a signed graph. That is, we label each edge of the graph with $1$ or $-1$...
4 votes
1 answer
313 views

Spectra of the quotient of a directed graph

Given a graph $G(V,E)$ and a partition $\{V_1,\dots V_n\}$ of the nodes set $V$, the adjacency and Laplacian spectra of the quotient graph $Q(G)$ interlaces the adjacent and the Laplacian spectra of ...
3 votes
0 answers
279 views

Cayley Graphs and Cyclically reduced words [closed]

Let $G$ be a finite group and $S$ be a symmetric generating set for $G$. (EDIT: Assume $S$ does not contain involutions!) Cyclically reduced words can be thought of as minimal length representatives ...
0 votes
0 answers
50 views

Intersection of ideals corresponding to simplicial complexes at different points?

Suppose I have two simplicial complexies $\triangle_1$ and $\triangle_2$. Consider their Stanley-Reisner ideals $I(\triangle_1)$ and $I(\triangle_2)$. I want to get their intersections when they meet ...
0 votes
1 answer
249 views

Find the minimum distance of some bad binary code

Let $M$ be a $n \times n$ matrix over the finite field of two elements that satisfies the following property$\colon$ the total number of 1's in each row coincides with one in each column. In other ...