Abstract
Given a finite set S and integer k≥0, let \(\binom{S}{k}\) denote the set of k-element subsets of S. A multiset may be regarded, somewhat informally, as a set with repeated elements, such as {1,1,3,4,4,4,6,}. We are only concerned with how many times each element occurs and not on any ordering of the elements. Thus for instance {2,1,2,4,1,2} and {1,1,2,2,2,4} are the same multiset: they each contain two 1’s, three 2’s, and one 4 (and no other elements).
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Notes
- 1.
We can apply the binomial theorem in this situation because I and Jcommute. If A and B are p ×p matrices that don’t necessarily commute, then the best we can say is \({(A + B)}^{2} = {A}^{2} + AB + BA + {B}^{2}\) and similarly for higher powers.
- 2.
All citations to the literature refer to the bibliography beginning on page 25.
References
A.E. Brouwer, W.H. Haemers, Spectra of Graphs (Springer, New York, 2012)
D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Applications, 3rd edn. (Johann Ambrosius Barth, Heidelberg/Leipzig, 1995)
D.M. Cvetković, P. Rowlinson, S. Simić, in An Introduction to the Theory of Graph Spectra. London Mathematical Society. Student Texts, vol. 75 (Cambridge University Press, Cambridge, 2010)
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Stanley, R.P. (2013). Walks in Graphs. In: Algebraic Combinatorics. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6998-8_1
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DOI: https://doi.org/10.1007/978-1-4614-6998-8_1
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