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2020-12-26 16:57:22 +0100 asked a question Consider the class of all possible connected simple graphs on $n$ vertices

Consider the class of all possible connected simple graphs on $n$ vertices ($n$ is any natural number, we can choose any natural number). Now from this collection, can we find those graphs (if there exists any) satisfying the following property: Suppose that $A$ denotes the usual $(0,1)$ adjacency matrix of a graph. Now can we find those graphs explicitly (if there is any) on $n$ vertices such that if $\lambda$ is an eigenvalue of $A$, then $\dfrac{1}{\lambda}$ is also an eigenvalue and if $\alpha$ is another eigenvalue (distinct from $\lambda$), then $-\dfrac{1}{\alpha}$ is an eigenvalue. Basically eigenvalues of $A$ are of the form $(\lambda,\dfrac{1}{\lambda})$ and $(\alpha,-\dfrac{1}{\alpha})$.

Basically I am trying to find the graphs for which some roots of the form $(\lambda,\dfrac{1}{\lambda})$, and some roots of the form $(\alpha,-\dfrac{1}{\alpha})$.

please help regarding this problem

2020-12-24 17:10:25 +0100 commented answer Graphs on 6 vertices with eigenvalue condition

This is a very nice answer. Thank you

2020-12-23 22:25:05 +0100 asked a question Graphs on 6 vertices with eigenvalue condition

Consider the class of all possible connected simple graphs on $6$ vertices. Now from this collection, can we find those graphs (if there exists any) satisfying the following property: Suppose that $A$ denotes the usual $(0,1)$ adjacency matrix of a graph. Now can we find those graphs explicitly (if there is any) on $6$ vertices such that if $\lambda$ is an eigenvalue of $A$, then $\dfrac{1}{\lambda}$ and $-\dfrac{1}{\lambda}$ both are eigenvalues. Simply the question can be said as: Characterize all possible simple connected graphs on $6$ vertices such that if $\lambda$ is an eigenvalue of $A$, then $\dfrac{1}{\lambda}$ and $-\dfrac{1}{\lambda}$ both are eigenvalues.

please help regarding this problem.

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2020-06-25 00:57:48 +0100 marked best answer Graphs having highest second smallest laplacian eigen value from a collection

Using this code

for G in graphs(7):
    if G.girth()==4:
        L = G.laplacian_matrix().eigenvalues()
        L.sort()
        show(L)
        G.show()

I have generated all graphs on $7$ vertices having girth=4. Now, from this code can we get the only unique graph having largest algebraic connectivity among all others.

2020-06-24 23:21:59 +0100 commented answer How to obtain the graph having highest algebraic connectivity?

Now, if we run the same coder for 8 vertices and 9 edges, we should get three graphs having algebraic connectivity=1. But I am getting only 2. Where is the fault. will you please help?

2020-06-24 23:10:56 +0100 commented answer How to obtain the graph having highest algebraic connectivity?

ok. I Have got it now. Thank you so much.

2020-06-24 23:00:23 +0100 commented answer How to obtain the graph having highest algebraic connectivity?

If possible, would you please give the whole code here so that I can obtain the two graphs. Thank you.

2020-06-24 22:47:50 +0100 commented answer How to obtain the graph having highest algebraic connectivity?

Ok. Thanks a lot for your answer.

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2020-06-24 22:24:16 +0100 asked a question How to obtain the graph having highest algebraic connectivity?

for G in graphs.nauty_geng("10-c"):

if G.size()==11:

L = G.laplacian_matrix().eigenvalues()

L.sort()

show(L)

G.show()

Using this code I have obtained the connected graphs on 10 vertices with 11 edges. Also I have obtained the Laplacian eigenvalues also for the corresponding graphs. The second smallest eigenvalue of Laplacian matrix is called the algebraic connectivity. Now among all these graphs, I need those graphs which have highest algebraic connectivity among all the graphs on 10 vertices with 11 edges. How we can do that?

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2020-03-25 16:03:27 +0100 commented question How to find an example of a 6 by 6 symmetric matrix (which is not identity matrix and does not satisfy $A^2=\text{ Identity matrix}$) which is similar to its inverse?

yes. I need a code that can give a matrix A with the property that A and $A^{-1}$ are similar over real field. Can I create some example of matrices in sage math which checks that whether A and $A^{-1}$ are similar over real field.. The obvious examples are easy for example a diagonal matrix. But can I create an example of a matrix so that A and $A^{-1}$ are similar over $\mathbb{R}$??

2020-03-25 08:47:23 +0100 commented question How to find an example of a 6 by 6 symmetric matrix (which is not identity matrix and does not satisfy $A^2=\text{ Identity matrix}$) which is similar to its inverse?

ok.can we have at least one such example?

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2020-03-24 21:43:18 +0100 asked a question How to find an example of a 6 by 6 symmetric matrix (which is not identity matrix and does not satisfy $A^2=\text{ Identity matrix}$) which is similar to its inverse?

How to find an example of a 6 by 6 symmetric matrix (which is not identity matrix and does not satisfy $A^2=\text{ Identity matrix}$)which is similar to its inverse? I am trying to find one such example in Sage. Please give some idea to proceed.

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2020-02-05 14:15:10 +0100 edited question Weighted adjacency matrix of a graph.

The code:

G=Graph(sparse=True, weighted=True)
G.add_edges([(0, 1, i), (0,8,1),(1,2,1),(2,3,1),(3,4,1),(4,5,1),(4,6,1),(6,7,1),(6,8,1),(8,9,1)])
M = G.weighted_adjacency_matrix()
show(M)

Here the (1,0) entry in the output matrix will be i, but I need -i. For that purpose, even I had defined (1,0,-i), it is not coming. Please give some hint.