# Spectra of Graphs

In Spring 2006, Andries E. Brouwer and Willem H. Haemers gave a
series of lectures at IPM, the Institute for Studies in Theoretical
Physics and Mathematics in Tehran. The lecture notes were combined
and published as an IPM report. Since that time, various versions
of that text have been available at this site.
Preprint PDF.

A book version was released by Springer on the 16th of December 2011.
However, the copyright year is 2012.

A.E. Brouwer & W.H. Haemers,
*Spectra of graphs*,
Springer,
New York, etc., 2012.

ISBN 978-1-4614-1938-9.

## Book Errata

p. 39, in the proof of Theorem 3.5.1, ‘θ_{n−α−1}’
should be ‘θ_{n−α+1}’.
p. 40, Proposition 3.6.3: add in part (ii) the condition
‘θ_{m} ≥ 0’, and in part (iii) the condition
‘θ_{t+m−1} ≥ 0’.

p. 55, 3rd line following Step 4: *semibipartite* should be *split*.

p. 57, middle: replace ‘N_{i}^{T}t_{0}...t_{i}’
by ‘N_{i+1}^{T}t_{0}...t_{i}’.

p. 87, Proposition 5.3.2, 2nd line: replace *eigenvalue* by
*eigenvector*.

p. 93, line 9: replace *edge-transitive* by *arc-transitive*.

p. 119, Proposition 9.1.4: add the word ‘primitive’:
*A primitive strongly regular graph ...*.

p. 117, 151 replace in the definition of ‘restricted eigenvalue’
the part *perpendicular to* by *which is not a multiple of*.

p. 125, line −9: replace ‘*K*_{5}’ by ‘*C*_{5}’.

p. 138, line 13, ‘*f* : *F→K*’ should be
‘*f* : *K→F*’.

p. 140, lines 13, 18: replace ‘*q*^{k−2}’
and ‘*q*^{k}’ by ‘*q*^{m−2}’
and ‘*q*^{m}’.

p. 155, line 9: replace ‘1’ by ‘≥1’ in the table entries for 126 and 176.

p. 160, bottom line, replace ‘(196,±1)’ by ‘(196,1)’.

p. 161, line 2: add in (iii) the condition ‘4|*a*’.

p. 166: in (iv) replace ‘*i*,*j*,*k*’
by ‘*i*,*j*’.

p. 167: in the line before Theorem 11.2.1, replace ‘*E*_{k}’
by ‘*E*_{i}’.

p. 167, Theorem 11.2.2: replace ‘For *i*’ by ‘For *j*’.

p. 169: in the line before Theorem 11.3.2,
replace ‘*b* × (*d*+1)’ by ‘*n* × (*d*+1)’.

p. 205, bottom line, instead of ‘*n*=6’, read ‘*n*=4’.

## Additions

On p. 37, line 2 a conjecture (by Nikiforov) on the sum of the spectral
radii of a graph and its complement is mentioned. This conjecture has
now been proved by Tamás Terpai.
On p. 210 it says: *One might wonder whether the disjoint union of
regular DS graphs with the same degree is always DS*.
Wonder no more! Both
2K_{3,3} + Σ ⊗ K_{2} and
3(C_{6} □ K_{2})
are bipartite cubic graphs with spectrum
±3^{3} ±2^{6} ±1^{3} 0^{12}.

On p. 163 a table with bounds for the maximum number of equiangular lines
in **R**^{d} is given. These bounds were taken from a survey
by Seidel, but no proofs are known, and perhaps they are lower bounds only.
Improved table, including recent results of
Barg-Yu, Azarija-Marc, Greaves-Koolen-Munemasa-Szöllősi, Greaves and Szöllősi:

1 |
2 |
3 |
4 |
5 |
6 |
7-13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23-41 |
42 |
43 |

1 |
3 |
6 |
6 |
10 |
16 |
28 |
28-29 |
36 |
40-41 |
48-50 |
54-60 |
72-75 |
90-95 |
126 |
176 |
276 |
276-288 |
344 |

## Thanks

Thanks to Sebastian Cioabă, Nathann Cohen, Péter Csikvári, Alexander Gavrilyuk,
Marko Orel, Dima Pasechnik and Ferenc Szöllősi for pointing out problems.