Univ. Beograd. Publ. Elektrotehn. Fak.

Ser. Mat. 10 (1999), 95{105.

THERE ARE EXACTLY 150 CONNECTED

INTEGRAL GRAPHS UP TO 10 VERTICES

Krystyna Bali�nska, Drago�s Cvetkovi�c, Mirko Lepovi�c, Slobodan Simi�c

A graph is called integral if its spectrum consists entirely of integers. Using

existing graph catalogues we established that there are exactly 150 connected

integral graphs up to 10 vertices. Adjacency matrices and/or pictures, and spectra

of these graphs together with some comments are given in this paper.

The spectrum of a graph is the spectrum of its (0; 1) adjacency matrix. Agraph G is called integral if its spectrum consists only of integers.

By the aid of computer we establish that there are exactly 150 connectedintegral graphs up to 10 vertices. In Section 1 we present data on graphs while inSection 2 we give some comments.

1. DATA ON INTEGRAL GRAPHS

Integral graphs up to 7 vertices can easily be found in published tables ofgraph spectra [3, 4, 6]. We have recently generated by a computer the �les of allnon-isomorphic graphs with 8; 9 and 10 vertices [8]. Using these �les, we easilyobtain all connected non-isomorphic integral graphs of order n = 8; 9; 10. Indepen-dently, these graphs have been generated in [1].

The numbers in of connected integral graphs with n vertices are given forn = 1; 2; : : : ; 10 in the following table:

n 1 2 3 4 5 6 7 8 9 10

in 1 1 1 2 3 6 7 22 24 83

Table 1.

Connected integral graphs with up to 5 vertices are easily obtained from thetables of spectra of connected graphs up to 5 vertices of [4]. They are K1, K2, K3,K4, C4, K5, 2K1 [K3 and K1;4.

The six connected integral graphs on 6 vertices have identi�cation numbers1; 9; 51; 52; 106 and 109 in the table of connected graphs with 6 vertices of [6]. They

1991 Mathematics Subject Classi�cation: 05C50

96 Krystyna Bali�nska, Drago�s Cvetkovi�c, Mirko Lepovi�c, Slobodan Simi�c

are K6, cocktail party graph CP (3), the graph of 3-sided prism C3 +K2, C6 andK2 � 2K1 (+ and � stand for graph sum (NEPS with basis f(0; 1); (1; 0)g - see, e.g.,[4], pp. 65-66) and corona).

There are 7 connected integral graphs on 7 vertices with identi�cation num-bers 4; 191; 235; 300; 710; 841 and 853 in the table of connected graphs on 7 verticesof [3]. They are S(K1;3), K153K2, L(K1;2+K2), L(K3;3 � e), C3 [ C4, C4 [ 3K1

and K7 (S, L and 5 stand for graph subdivision, line graph and graph join, re-spectively).

Integral graphs with 8, 9 and 10 vertices are displayed in Lists 1, 2 and 3,respectively. Graphs in the lists are represented in the form:

n1 a12 a13 a23 : : : a1n a2n : : : an�1;n ;

where n1 is the identi�cation number of the corresponding integral graph, anda12 a13 a23 : : : a1n a2n : : : an�1;n is the upper diagonal part of its adjacency ma-trix. For each graph we have chosen the adjacency matrix which corresponds tolexicographical ordering of vertices with respect to vertex angle sequences [7].

The integral graphs with 8; 9 and 10 vertices are ordered lexicographically bytheir eigenvalues in non-increasing order; eigenvalues are given in Lists 4; 5, and 6,respectively. In these lists, a multiple eigenvalue is not repeated and its multiplicityis given in the form of an exponent. In the sets of cospectral integral graphs the(common) spectrum is given only once. Cospectral graphs are ordered by meansof their angles [7].

LIST 1. ADJACENCY MATRICES OF INTEGRAL GRAPHS WITH 8

VERTICES

1. 1 11 111 1111 11111 111111 1111111

2. 1 10 011 1111 11110 111111 1111110

3. 1 11 110 1110 11010 111010 1101010

4. 1 10 110 1010 01111 111110 0111110

5. 1 11 110 1110 11011 111000 1101001

6. 1 11 111 1111 11110 101000 0101000

7. 1 11 111 1110 11100 111000 1110000

8. 1 10 010 1010 01110 101110 0101110

9. 1 11 110 1101 10100 011001 0001100

10. 0 11 110 1100 11100 110100 1100100

11. 1 11 110 1110 11010 100000 0100001

12. 1 11 111 1011 11000 000011 0100000

13. 1 11 111 1001 01010 101000 0110000

14. 1 11 111 1110 01100 100000 1000000

15. 1 10 100 1000 01111 011110 0111100

16. 1 11 110 1100 11000 110000 1100000

17. 0 11 111 0100 10000 010010 1000010

18. 1 10 010 0010 00011 100110 0110010

19. 0 11 111 0100 10000 100010 0100010

20. 1 11 110 1000 01001 001000 0001000

21. 0 11 110 1100 11000 010000 1000000

22. 1 11 110 1000 01000 100000 0100000

There are exactly 150 connected integral graphs up to 10 vertices 97

LIST 2. ADJACENCY MATRICES OF INTEGRAL

GRAPHS WITH 9 VERTICES

1. 1 11 111 1111 11111 111111 1111111 11111111

2. 1 11 111 1110 11010 111011 1101110 11110000

3. 1 11 111 1110 10111 110111 1111000 11101000

4. 1 10 101 0110 01011 111111 1111110 11111100

5. 1 11 111 1110 11100 110111 1101110 11110000

6. 1 11 110 1100 10111 101110 0111111 01111110

7. 1 11 111 1111 11000 101001 0110011 00011000

8. 1 11 110 1100 10111 101110 1101000 11100000

9. 1 00 001 1111 11110 111100 1111000 11110000

10. 1 00 001 1100 00110 100111 0110110 11110000

11. 1 00 001 0011 11001 110010 0011011 11110000

12. 1 00 101 1011 01110 011011 1100000 11000000

13. 1 11 011 0111 10100 110001 1000000 10000001

14. 1 00 101 0111 01110 101011 1100000 11000000

15. 1 11 110 1011 10000 100001 1000010 10000011

16. 1 10 011 1111 10100 101000 0101000 01010000

17. 1 11 110 0011 00111 110000 1100000 11000000

18. 1 11 100 1001 01010 010011 0011010 00101011

19. 1 11 100 0010 01000 010111 0011110 10011100

20. 1 11 110 1011 01010 001011 0110000 10000000

21. 1 11 101 1101 00101 010101 0110000 10000000

22. 0 01 111 1111 11000 101000 1010000 11000000

23. 1 11 110 1100 10100 101000 0110000 01100000

24. 1 00 001 1011 01110 110000 1000000 01000000

LIST 3. ADJACENCY MATRICES OF INTEGRAL

GRAPHS WITH 10 VERTICES

1. 1 11 111 1111 11111 111111 1111111 11111111 111111111

2. 1 11 111 1111 11110 111011 1101111 10111111 011111111

3. 1 11 111 1111 11111 111111 1111110 11111100 111111000

4. 1 00 001 1111 11111 111110 1111101 11110110 111101011

5. 1 11 111 1111 11111 111111 1110100 11010010 101101000

6. 1 11 111 1111 11111 111111 1111000 11001100 001111000

7. 1 11 111 1111 11111 111100 1111001 11110000 111100001

8. 1 00 001 1111 11110 111111 1111110 11110011 111111001

9. 1 11 111 1100 11001 001101 0011101 00111111 110011110

10. 1 11 111 1101 11011 111010 1110011 00011100 001000111

11. 1 11 111 1110 11101 111100 1100110 11010010 100100010

12. 1 11 111 1111 11110 111011 0110000 11000001 101000011

13. 1 11 000 0001 00011 111111 1111110 11111100 111111000

14. 1 11 111 1111 11111 111000 1110000 00011100 000111000

15. 1 11 110 1101 10110 101011 0111010 01101011 000111111

16. 1 00 110 1011 01110 111011 1110001 10111101 011111010

17. 1 11 111 1111 11111 111000 1001010 01001100 001110000

18. 1 10 010 0010 00011 111111 1111110 11111100 111111000

19. 1 00 001 1111 11111 111111 1111000 11110000 111100000

20. 1 11 111 1111 11111 011110 1001100 11100000 100000000

21. 1 11 111 1110 11100 110111 1101110 00011100 001000111

22. 1 11 111 1111 11110 111100 1111000 11000000 001100000

23. 1 10 011 1111 11110 111100 1111000 11110000 111100000

24. 1 11 111 1111 11110 000010 0000101 00000111 000001111

25. 1 10 101 1011 10111 011000 0101001 01001011 010001111

26. 0 11 111 1100 11001 110011 1111000 00110001 000011100

98 Krystyna Bali�nska, Drago�s Cvetkovi�c, Mirko Lepovi�c, Slobodan Simi�c

27. 1 11 111 1100 11001 101100 1000110 01100100 010110000

28. 0 11 110 1111 11111 011000 1010000 10010001 010100100

29. 1 11 111 1100 10101 100111 0111000 00001111 100000000

30. 1 11 110 1101 10110 101011 0001111 01100000 011000000

31. 1 11 111 1111 11100 100110 0000011 00000110 100000011

32. 1 11 110 1101 11100 101100 1011000 01001100 010011000

33. 1 10 011 1111 11111 100100 1001001 01100000 011000001

34. 1 11 110 1100 10111 101110 0001111 01100000 011000000

35. 1 11 111 1100 11001 101010 1001010 01100100 010110000

36. 1 10 011 1111 11111 110000 1010000 00110010 010100010

37. 0 11 110 1110 11010 110011 1111000 00001101 001100100

38. 1 10 011 1111 11111 110000 0011001 10010000 011000001

39. 1 11 111 1001 10010 100100 0110110 01101010 011001100

40. 1 11 111 1100 11001 110000 1100001 11000000 110000001

41. 1 11 111 1111 10110 011100 1100000 10000000 010000000

42. 0 11 111 1111 11111 110000 1100000 10000000 010000000

43. 1 11 110 1100 10111 101110 0100011 00111001 100000000

44. 1 10 100 1000 10000 011111 0111110 01111100 011111000

45. 1 11 111 1100 11001 000100 0010001 00000011 000000111

46. 1 11 101 1101 00011 111000 0000010 00000101 000001011

47. 1 11 110 1100 11000 000101 0010100 00100101 000110100

48. 0 11 110 1001 10101 010100 0110001 11000000 001100001

49. 1 10 011 1100 11000 001101 1100000 00110001 001110000

50. 1 10 011 1001 10011 011000 0110001 11000000 001100001

51. 1 11 111 1000 10001 100011 0100001 00101000 000101000

52. 1 10 110 1010 10011 111000 0101000 00101000 000001100

53. 1 11 111 0100 11001 000110 0010101 10000000 100000001

54. 1 11 101 1101 00011 011001 0000011 10000000 100000001

55. 1 11 110 1100 11000 001100 0000111 11000000 000000110

56. 0 10 101 1110 11100 110100 1101000 01000000 010000001

57. 1 11 110 1011 01001 001101 1000000 10000001 000001100

58. 1 00 001 1111 11110 110000 0011000 10100000 010100000

59. 0 10 011 0111 10111 110000 1100000 11000000 110000000

60. 1 11 111 1100 11001 000100 0010001 00001000 000001001

61. 1 11 100 1101 10110 000111 0110000 01000000 001000000

62. 1 11 110 1100 11000 001010 0011000 00001101 000101100

63. 0 11 110 1010 10010 011001 0101100 11000000 001100001

64. 1 11 110 1010 10010 100011 0100001 00100100 000110000

65. 1 10 100 1000 01110 011010 0101100 00111000 000001111

66. 0 10 010 1111 11110 100100 1100000 01100000 001100000

67. 0 11 110 1100 00111 110000 1100000 00110010 001100010

68. 0 11 110 1100 11000 001110 0011010 00101100 110000000

69. 0 00 111 1110 11100 111000 1100000 10100000 011000000

70. 1 11 111 1100 10100 011000 1000000 01000000 001000000

71. 0 11 110 1100 11000 110000 1100000 11000000 110000000

72. 0 10 010 1010 01010 001100 0000110 00000011 110000001

73. 0 00 000 1100 00111 001100 0011000 11000010 110000010

74. 1 00 001 1100 00110 011000 0101000 10010000 101000000

75. 0 11 110 1100 11000 001000 0001000 00000011 000000110

76. 1 00 001 1010 01010 010100 0101000 10100000 101000000

77. 1 11 100 1001 01000 001001 0001000 00001001 100000000

78. 1 11 100 1001 00011 010000 0010000 00100000 010000000

79. 1 10 100 0100 01000 001010 0010010 00011001 000101100

80. 1 10 100 1000 10000 010001 0110000 01010000 010010000

81. 0 11 110 1100 11000 000001 0010000 00010000 000010000

82. 1 10 011 1001 01100 100000 0100000 00100000 000100000

83. 1 10 100 1000 10000 100000 1000000 10000000 100000000

There are exactly 150 connected integral graphs up to 10 vertices 99

LIST 4. SPECTRA OF INTEGRAL GRAPHS WITH 8 VERTICES

1. 7 �17

2. 6 04 �23

3. 5 2 �15 �2

4. 5 12 �14 �3

5. 5 12 �13 �22

6. 5 1 02 �12 �22

7. 5 04 �12 �3

8. 4 2 03 �23

9. 4 2 02 �12 �22

10. 4 12 0 �13 �3

11-13. 4 12 0 �12 �22

14. 4 1 03 �1 �22

15. 4 06 �4

16. 4 05 �1 �3

17. 3 2 1 �14 �2

18. 3 13 �13 �3

19. 3 13 �12 �22

20. 3 12 02 �1 �22

21. 3 1 04 �1 �3

22. 3 1 04 �22

LIST 5. SPECTRA OF INTEGRAL GRAPHS WITH 9 VERTICES

1. 8 �18

2-3. 6 12 0 �12 �23

4. 6 1 04 �22 �3

5. 6 1 03 �12 �2 �3

6. 6 06 �32

7. 5 2 03 �1 �23

8. 5 12 02 �12 �2 �3

9. 5 1 04 �12 �4

10. 4 2 12 �12 �23

11. 4 2 1 02 �12 �2 �3

12. 4 2 1 02 �1 �23

13. 4 2 1 0 �13 �22

14. 4 2 04 �1 �2 �3

15-16. 4 2 04 �23

17. 4 2 03 �13 �3

18. 4 14 �24

19. 4 13 02 �22 �3

20. 4 13 0 �1 �23

21-22. 4 12 03 �1 �2 �3

23. 4 12 03 �23

24. 3 2 1 02 �12 �22

100 Krystyna Bali�nska, Drago�s Cvetkovi�c, Mirko Lepovi�c, Slobodan Simi�c

LIST 6. SPECTRA OF INTEGRAL GRAPHS WITH 10 VERTICES

1. 9 �19

2. 8 05 �24

3. 8 03 �15 �3

4. 7 12 02 �12 �22 �3

5-6. 7 12 0 �13 �23

7. 7 12 �16 �3

8. 7 1 04 �12 �32

9. 6 2 1 02 �12 �22 �3

10. 6 2 1 02 �1 �24

11. 6 2 1 0 �13 �23

12. 6 2 04 �24

13. 6 2 03 �14 �4

14. 6 2 02 �15 �3

15. 6 14 �25

16. 6 13 02 �23 �3

17. 6 13 �13 �23

18. 6 12 03 �12 �2 �4

19. 6 12 02 �14 �4

20. 6 12 02 �12 �23

21. 6 1 05 �1 �2 �4

22. 6 1 04 �12 �2 �3

23. 6 07 �2 �4

24. 5 3 1 �15 �22

25. 5 3 04 �24

26. 5 3 02 �15 �3

27-28. 5 2 12 �13 �23

29-30. 5 2 1 03 �24

31-33. 5 2 1 02 �13 �2 �3

34. 5 2 05 �22 �3

35-36. 5 14 �1 �24

37. 5 13 02 �12 �32

38. 5 13 02 �1 �22 �3

39. 5 13 0 �14 �4

40. 5 13 �15 �3

41. 5 12 03 �1 �23

42. 5 12 02 �14 �3

43. 5 1 06 �2 �4

44. 5 08 �5

45-46. 4 3 1 0 �14 �22

47-50. 4 2 12 0 �13 �2 �3

51-54. 4 2 12 0 �12 �23

55-59. 4 2 1 03 �12 �2 �3

60-61. 4 2 1 03 �1 �23

62-63. 4 14 0 �1 �22 �3

64. 4 14 0 �24

65. 4 14 �14 �4

66. 4 13 03 �22 �3

There are exactly 150 connected integral graphs up to 10 vertices 101

67-68. 4 13 02 �13 �4

69. 4 12 04 �12 �4

70. 4 12 04 �23

71. 4 08 �4

72. 3 2 13 �12 �23

73. 3 2 12 02 �12 �2 �3

74. 3 2 12 02 �1 �23

75-76. 3 2 1 04 �1 �2 �3

77. 3 2 1 04 �23

78. 3 2 1 03 �12 �22

79. 3 15 �24

80. 3 14 �14 �3

81-82. 3 13 02 �13 �3

83. 3 08 �3

Integral graphs with 8 and 9 vertices are depicted in Figures 1 and 2, respec-tively.

2. SOME COMMENTS

Based on the data on integral graphs presented in this paper several observa-tions can be made. We shall �rst comment some facts concerning sets of cospectralintegral graphs.

No two non-isomorphic connected integral graphs on less than 8 vertices arecospectral. There are just one triplet of connected integral graphs on 8 vertices,3 pairs on 9 vertices and 10 pairs, 1 triplet, 2 quadruples and 1 quintuple on 10vertices. All graphs contained in these sets are non-regular.

The above cospectral graphs can be distingushed by angles. Indeed, smallestcospectral graphs with the same angles have 10 vertices [5] and among 58 pairs ofsuch cospectral graphs no one has an integral spectrum.

There are integral cospectral graphs on less than 8 vertices but then at leastone of them is disconnected. These cases include the smallest pair of cospectralgraphs: K1;4; C4 [K1. The common spectrum is 2; 03;�2. It is also known thatintegral graph S(K1;3) has a cospectral-mate in the disconnected graph C6 [ K1

with the spectrum 2; 12; 0;�12;�2. An interesting case of two disconnected integralgraphs has been noted in [2]. Graphs in question have 20 vertices. One consists oftwo components equal to graph No. 73 of List 6 and the components of the otherone are the graph of a hexagonal prism (C6+K2) on 12 vertices and the cube graph(C4 +K2) on 8 vertices (graph No. 18 of List 4).

Some of cospectral graphs have the same main angles. This happens in thefollowing pairs of cospectral graphs: 2 and 3 on 9 vertices, 5 and 6, 45 and 46, 51 and52, 53 and 54, 55 and 56, 67 and 68 on 10 vertices, as well as in the triplet 48, 49 and50 on 10 vertices. Note that main angles are not mutually equal in all four graphsfrom the cospectral quadruples 47-50 and 51-54. The complements of cospectralgraphs sharing the same main angles are cospectral. Complements of graphs fromcospectral quadruple 51-54 form two pairs of cospectral graphs but not a cospectral

102 Krystyna Bali�nska, Drago�s Cvetkovi�c, Mirko Lepovi�c, Slobodan Simi�c

Fig. 1.

There are exactly 150 connected integral graphs up to 10 vertices 103

Fig. 2.

104 Krystyna Bali�nska, Drago�s Cvetkovi�c, Mirko Lepovi�c, Slobodan Simi�c

quadruple. Similarly, the complements of graphs 47-50 do not provide a cospectralquadruple; we get in this case only a cospectral triplet (complements of graphs48, 49 and 50). However, among graphs of all these sets of cospectral graphs onlycomplements of graphs 2 and 3 on 9 vertices are integral. Both complements aredisconnected (the �rst has three components two of them being isolated vertices,while the second has two components one being an isolated vertex) and thereforedo not appear in List 2. Cospectral integral graphs with cospectral integral com-plements were noted for the �rst time in [9], where an example with 11 vertices isgiven. It is interesting that we have the same situation with isolated vertices asabove: adding two isolated vertices to graph 18 on 9 vertices (List 5) and addingan isolated vertex to graph 64 on 10 vertices (List 6) produce the example from [9].

However, several other graphs of our lists have integral complements. If thecomplement is connected, it is included in the lists. In this way we can identify thefollowing pairs of complementary integral graphs: 8 and 18 on 8 vertices, 10 and19 on 9 vertices, 9 and 73, 15 and 79, 16 and 72, 25 and 65 on 10 vertices. Thesmallest such pair is with 6 vertices: C6 and C3+K2. All these graphs are regular.Selfcomplementary graphs appear in our lists as well: graph 13 on 8 vertices (non-regular) and graphs 11 and 18 on 9 vertices (both regular). The following graphshave integral disconnected complements: 1-5, 7, 10, 15, 16 on 8 vertices, 1-6, 8, 9,15 on 9 vertices and 1-4, 7, 8, 13, 18, 23, 40, 44, 71, 83 on 10 vertices.

Several other observations and data on integral graphs up to 10 vertices arecontained in [1]. In particular, the number of automorphisms is given for eachgraph.

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There are exactly 150 connected integral graphs up to 10 vertices 105

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24A(1987), 21{27.

The Technical University of Pozna�n, (Received May 19, 1999)

pl. M. Sklodowskiej-Curie 5,

60-965 Pozna�n, Poland

kbal@agat.sk-kari.put.poznan.pl

University of Belgrade,

Faculty of Electrical Engineering,

P.O. Box 35-54, 11120 Belgrade,

Yugoslavia

ecvetkod@ubbg.etf.bg.ac.yu

esimics@ubbg.etf.bg.ac.yu

Tihomira Vuksanovi�ca 32,

34000 Kragujevac, Yugoslavia

lepovic@uis0.uis.kg.ac.yu