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        3
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        0answers
        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 ...
        1
        vote
        0answers
        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 ...
        0
        votes
        1answer
        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, ...
        9
        votes
        0answers
        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 ...
        5
        votes
        1answer
        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 $$\prod_{uv\in E}(x_{\sigma(u)}-x_{\sigma(v)})=-\prod_{uv\in E}(x_u-x_v)?$$ ...
        2
        votes
        1answer
        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 ...
        5
        votes
        1answer
        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 ...
        0
        votes
        1answer
        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 ...
        1
        vote
        0answers
        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 ...
        5
        votes
        1answer
        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 ...
        4
        votes
        1answer
        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 ...
        4
        votes
        0answers
        170 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)...
        1
        vote
        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 ...
        1
        vote
        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....
        1
        vote
        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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