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Preliminaries, a partitioned matrix and Cauchy–Binet formula. Important properties of eigenvalues of symmetric matrices, including Spectral Theorem and Interlacing Theorem are reviewed. Basic notions of generalized inverses, including Moore–Penrose inverse are stated. Relevant concepts and results are given, although we omit the proofs.

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Regular Graphs,the line graph of a tree and those of the Laplacian as well as the signless Laplacian are shown to be related to each other. This connection is exploited to obtain several statements centered around the classical result that the nullity of the line graph of a tree is at most one.

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Wolfgang Scholl,Frank Schmelzer,Sandra Tirre Stanley on counting directed paths is proved. In the final section we discuss trees which have a nonsingular adjacency matrix, and identify the cases when the inverse of the adjacency matrix corresponds to a graph.

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https://doi.org/10.1007/978-3-658-17121-6djacency matrix of a regular graph and that of its complement and line graph. Several results in this direction are proved in the next section. In the final section we derive spectral properties of strongly regular graph and apply them to derive the well-known Friendship Theorem.

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