# 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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### 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 $\Sigma$ ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
• 351
1 vote
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### 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
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### How many line graphs are there?

I am thinking of a quantitative (possibly based on random graph theory) or qualitative (say, based on topological ideas, e.g. Baire's theorem in the Gromov-Hausdorff metric space) information about ...
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