Abstract
A graph is called a block graph if each of its blocks is a complete graph. In the first section we obtain a formula for the determinant of the adjacency matrix of a block graph. The line graph of a tree is a block graph. Some basic properties of the signless Laplacian are proved. The eigenvalues of 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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References and Further Reading
Bapat, R.B., Raghavan, T.E.S.: Encyclopedia of Mathematics and Its Applications. Nonnegative Matrices and Applications. Cambridge University Press, Cambridge (1997)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9. SIAM, Philadelphia (1994)
Cameron, P., J.: Strongly regular graphs, In Selected Topics in Graph Theory L.W. Beineke and R.J. Wilson, Ed. Academic Press, New York, pp. 337–360 (1978).
Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall Inc, New York (1993)
Koolen, J.H., Moulton, V.: Maximal energy graphs. Adv. Appl. Math. 26, 47–52 (2001)
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© 2014 Springer-Verlag London
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Bapat, R.B. (2014). Regular Graphs. In: Graphs and Matrices. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-6569-9_6
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DOI: https://doi.org/10.1007/978-1-4471-6569-9_6
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