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76 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 ...
64 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 ...
91 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, ...
243 views

Correspondence between matrix multiplication and a graph operation of Lovasz

In his book "Large networks and graph limits" (available online here: http://web.cs.elte.hu/~lovasz/bookxx/hombook-almost.final.pdf), Lovasz describes a multiplication operation (he calls it ...
125 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 $$\prod_{uv\in E}(x_{\sigma(u)}-x_{\sigma(v)})=-\prod_{uv\in E}(x_u-x_v)?$$ ...
170 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 ...
162 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 ...
99 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 ...
35 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 ...
117 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 ...
314 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 ...
170 views

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 $\alpha(G)... 1answer 83 views Determinant of incidence matrix of a unicyclic unbalanced signed graph While reading a paper on unicyclic unbalanced signed graphs, I met the following fact: The determinant of the incidence matrix of a unicyclic unbalanced graph (i.e. the cycle of the graph has an ... 0answers 91 views graphs with semiregular automorphisms I need some "well-known" non-regular finite graphs (at least two vertices have different valency) whose automorphism groups contain a non-trivial subgroup that acts on the vertices semi-regularly (i.... 1answer 187 views Automorphism group of a graph Suppose$\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(\...

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