# A First Course in Graph Theory and CombinatoricsBlock Designs

A First Course in Graph Theory and Combinatorics: Block Designs [Let V be an n-dimensional vector space over the finite field Fq of q elements. We would like to determine the number of subspaces of dimension k. For example, the number of 1-dimensional subspaces is easily found as these are subspaces spanned by one element. Such an element must be non-zero and there are qn − 1 ways of choosing such an element. But for each choice, any non-zero scalar multiple of it will generate the same subspace as there are q − 1 such multiples for any fixed vector, we get a final tally of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{{q^{n - 1}}}}{{q - 1}}$$\end{document} for the number of 1-dimensional subspaces of V. This gives us a clue of how to determine the general formula.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A First Course in Graph Theory and CombinatoricsBlock Designs

Part of the Texts and Readings in Mathematics Book Series (volume 55)
Springer Journals — May 24, 2017
17 pages

/lp/springer-journals/a-first-course-in-graph-theory-and-combinatorics-block-designs-nMIGpkxTkm
Publisher
Hindustan Book Agency
© Hindustan Book Agency (India) 2009
ISBN
978-81-85931-98-2
Pages
100 –117
DOI
10.1007/978-93-86279-39-2_9
Publisher site
See Chapter on Publisher Site

### Abstract

[Let V be an n-dimensional vector space over the finite field Fq of q elements. We would like to determine the number of subspaces of dimension k. For example, the number of 1-dimensional subspaces is easily found as these are subspaces spanned by one element. Such an element must be non-zero and there are qn − 1 ways of choosing such an element. But for each choice, any non-zero scalar multiple of it will generate the same subspace as there are q − 1 such multiples for any fixed vector, we get a final tally of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{{q^{n - 1}}}}{{q - 1}}$$\end{document} for the number of 1-dimensional subspaces of V. This gives us a clue of how to determine the general formula.]

Published: May 24, 2017