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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 $G$.

<p>Also find the eigen values of $L(G\times H)$ in terms of $L(G)$ and $L(H)$.</p>

My try:

Let $|V(G)|=n;|V(H)|=m$ .Then $|V(G\times H)|=m\times n$.

Now two vertices $(v,w)\sim (v_1,w_1)\iff v\sim v_1 \text{and} w\sim w_1$.

I am unable to understand how the $(i,j)^{th}$ entry of $L(G\times H)$ will look.

Please give some hints. I am unable to solve the problem.

Learnmore
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  • Follow the paper - Barik, S., Bapat, R.B. and Pati, S., 2015. On the Laplacian spectra of product graphs. Applicable Analysis and Discrete Mathematics, 9(1), pp.39-58. – G_0_pi_i_e May 26 '16 at 02:57
  • @G_0_pi_i_e; I have just started reading spectral graph theory from the book "Graphs and Matrices" Laplacian Matrices ; it would have been better if you could give some hints on solving the problem – Learnmore May 26 '16 at 12:11
  • I don't have any experience on how to read papers – Learnmore May 26 '16 at 12:11
  • Anyway ;I found you are doing research in Spectral in IIT ;can you please give some books apart from Bapat which I can use /you are using as a beginner; – Learnmore May 26 '16 at 12:13
  • Books: 1. Algebraic Graph Theory by Godsil, Royle. 2. Spectra of graphs by Cvetkovic, Doob, Sachs. 3. Spectra of graphs by Brouwers and Haemers. – G_0_pi_i_e May 27 '16 at 02:39
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    For the problem: Do you know Kronecker product of matrices? If not first have some idea about it and some of its properties. Then by considering one example you can be sure that $$L(G\times H)=L(G)\otimes I_n+I_m\otimes L(F),$$ where $\otimes$ stands for Kronecker product operation. Then the further result becomes immediate. – G_0_pi_i_e May 27 '16 at 02:47
  • I am extremely grateful for your comments – Learnmore May 27 '16 at 06:18
  • But I am unable to do the proof of the given result ;though I have used an example ;I am unable to prove it formally – Learnmore May 27 '16 at 06:19
  • Will you please help; – Learnmore May 27 '16 at 06:19
  • Do you get its matrix representation? – G_0_pi_i_e May 28 '16 at 06:28

1 Answers1

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Important properties of Kronecker product operation: $$(A\otimes B)(C\otimes D)=AC\otimes BD,$$ $$A\otimes(B+C)=A\otimes B+A\otimes C.$$ Now if $L(G)X_i=\lambda_i X_i$ and $L(F)Y_j=\mu_j Y_j$, then \begin{align*}L(G\times H)(X_i\otimes Y_j)&=(L(G)\otimes I_m+I_n\otimes L(H))(X_i\otimes Y_j)\\&=L(G)X_i\otimes I_mY_j+I_nX_i\otimes L(H)Y_j\\&=\lambda_iX_i\otimes Y_j+\mu_jX_i\otimes Y_j\\&=(\lambda_i+\mu_j)X_i\otimes Y_j.\end{align*}

G_0_pi_i_e
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  • I have got all the steps ;my trouble is I am not getting why $L(G\times H)=L(G)\times I_m+I_n\times L(H)$ – Learnmore May 28 '16 at 10:42
  • It's not $L(G)\times I_m$, rather $L(G)\otimes I_m$. That is why I am telling you go through Kronecker product. https://en.wikipedia.org/wiki/Kronecker_product – G_0_pi_i_e May 30 '16 at 03:10
  • Take one example and write down the name of the vertices as $(v_G, v_H)$. Consider them in lexicographic order and try to form $A(G\times H)$, the adjacency matrix. I am sure you can get your result. – G_0_pi_i_e May 30 '16 at 03:15