# Tagged Questions

For questions related to the study of properties of a graph in relationship to the spectral properties of some associated matrix.

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### How to find the eigen values

How to find the eigen values of the graph having vertex set as $\{1,2,.......n\}$ and edge set as $\{(l,l+1)\}$ $\cup (1,n)$ ? where $1\le l \le n$. Here I am considering the Laplacian matrix of ...
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### Spectral Graph Theory :Cartesian product of Laplace Matrix

Let $G\times H$ be the Cartesian Product of $G$ and $H$. Determine $L(G\times H)$ in terms of $L(G)$ and $L(H)$ where $L(G)$ denotes Laplacian Matrix of $G$. Also find the eigen ...
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### How to read Spectral Theory of Graphs

My background is a course is Linear Algebra -Hoffman,Kunze Graph Theory-Frank Harary I am doing a coursework in Spectral Graph Theory . As I am going through it, I am searching for some ...
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### Prove Cycle Graphs are never hyper-energetic.

The energy of a graph G, is defined as $$\varepsilon(G)=\sum_{i=1}^{n}\left|\lambda_i\right|$$ where $\lambda_i$ are the eigenvalues associated with the adjacency matrix of the Graph. The energy of a ...
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### Number of regular tournaments

A regular tournament is a tournament where each player has the same number of wins. Since each player plays $n-1$ games, a regular tournament must have an odd number of players. My question is - 'how ...
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### Geometric intuition of graph Laplacian matrices

I am reading about Laplacian matrices for the first time and struggling to gain intuition as to why they are so useful. Could anyone provide insight as to the geometric significance of the Laplacian ...
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### How to prove the determinant of resistance matrix $R$?

Let $G$ be a connected graph with $n$ vertices, $R$ be the resistance matrix of $G$, $\tau$ be the $n\times 1$vector with components $\tau_1,\tau_2,...\tau_n$,and $\tau_i=2-\sum_{j\thicksim i} r(i,j)$ ...
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### Motivation for spectral graph theory.

Why do we care about eigenvalues of graphs? Of course, any novel question in mathematics is interesting, but there is an entire discipline of mathematics devoted to studying these eigenvalues, so ...
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### Connection between the Tutte and characteristic polynomials?

Both the Tutte polynomial $T_G(x,y)$ and the characteristic polynomial $\phi_G(x)$ encode a great amount of structure of the input graph $G$. I've read somewhere that the Tutte polynomial has a kind ...
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### Fiedler vector for weighted graphs

The second smallest eigenvalue of the Laplacian matrix of a (connected) graph is known as the algebraic connectivity of a graph and the corresponding eigenvectors are known as Fiedler vectors. I got ...
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### Algebraic Graph Theory: What is an integral eigenvalue?

I'm having some trouble with the an problem out of Bondy and Murty's Graph Theory (2008): 1.1.21 b) Show that rational eigenvalues of a graph are integral. I understand that this is a statement ...
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### Spectral radius of “almost” regular graph ?!

The answer to this question could be trivial. The Graph Let $G$ be graph formed of two $d$-regular connected components. That is, $G= H_1\cup H_2$, where $H_1$, and $H_2$ are $d$-regular and ...
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### Commuting weighted Laplacians

During my research I am concerned with the problem of finding commuting weighted Laplacian matrices (w.r.t finite simple weighted graphs on the same vertex sets). My question is about necessary ...
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### ELI5: What is spectral graph theory?

I am aware that there is already a similar question here, but unfortunately I find the discussion there to be beyond my grasp. I am looking for an intuitive explanation of spectral graph theory, as ...
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### nonisomorphic cospectral regular graphs

Is there any non-isomorphic cospectral regular graphs? Please suggest some examples or reference. Cospectral graphs are those graph which possess same set of spectrum.