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Is there a nice way of relating the adjacency, incidence , Laplacian matrices and other matrices associated to a graph of a total graph with its original graph, or, say, at least relating that of the line graph with the graph?

I think there must be some relation. I also think the rank of the Laplacian of the graph and that of the graph also are closely related. Are there any references for this? Thanks beforehand.

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If $C$ is the incidence matrix (rows indexed by vertices, columns by edges, two 1s in each column) then $CC^T$ is the Laplacian matrix and $C^TC-2I is the adjacency matrix of the linegraph. From this it follows that the non-zero eigenvalues of the Laplacian are the same as the non-zero (eigenvalues + 2) of the linegraph.

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Not sure if I really understand what exactly you are looking for. There is a paper by Vic Reiner and some of his former REU students on the “critical group of a line graph”. This relates the Smith normal form of the laplacian matrix to that of its line graph, in a pretty strong way. The Smith normal form gives you, in particular, the rank of a matrix over any field.

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  • $\begingroup$ The paper you refer to has good results, but unfortunately quite unrelated to the question I asked. I just asked what the relation(in mostly linear algebraic parameters) between the adjacency matrix/incidence matrix/laplacian matrix of a graph and the same matrices of its total graph is? If there is no known relation, my next question was whether such relation exists between a graph and its line graph $\endgroup$
    – vidyarthi
    Commented Apr 11, 2019 at 21:46

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