Univ. Beograd. Publ. Elektrotehn. Fak.
Ser. Mat. 9 (1998), 109–123.
4-REGULAR INTEGRAL GRAPHS
D. Cvetković, S. Simić, D. Stevanović
Possible spectra of 4-regular integral graphs are determined. Some constructions
and a list of 65 known connected 4-regular integral graphs are given.
1
Introduction
A graph is called integral if all its eigenvalues are integers. The quest for integral
graphs was initiated by F. Harary and A. J. Schwenk [12]. All such connected
cubic graphs were obtained by D. Cvetković and F. C. Bussemaker [6, 3],
and independently by A. J. Schwenk [15]. There are exactly thirteen connected
cubic integral graphs. In fact, D. Cvetković [6] proved that the set of connected
regular integral graphs of a fixed degree is finite. Similarly, the set of connected
integral graphs with bounded vertex degrees is finite. Z. Radosavljević and
S. Simić [16] determined all 13 connected nonregular nonbipartite integral graphs
whose maximum degree equals four. Recently, D. Stevanović [17] determined all
24 connected 4-regular integral graphs avoiding ±3 in the spectrum. In this paper
we are interested in connected 4-regular integral graphs.
The search for integral graphs becomes easier if we use product of graphs.
Given two graphs G and H, with vertex sets V (G) and V (H), their product G × H
is the graph with vertex set V (G) × V (H) in which two vertices (x, a) and (y, b)
are adjacent if and only if x is adjacent to y in G and a is adjacent to b in H. If
λ1 , . . . , λn are the eigenvalues of G, and µ1 , . . . , µm are the eigenvalues of H, then
the eigenvalues of G × H are λi µj for i = 1, . . . , n, j = 1, . . . , m (see [8]). If graph
G is connected, nonbipartite, 4-regular and integral, then the product G × K2 is
connected, bipartite, 4-regular and integral, since the eigenvalues of K2 are 1 and
−1. Therefore, in determining 4-regular integral graphs we can consider bipartite
graphs only, and later extract such nonbipartite graphs from the decompositions
of bipartite ones in the form G × K2 . On the other hand, if G is bipartite then
G × K2 = 2G and we cannot obtain new graphs by iterating product with K2 .
In Section 2 we find all possible spectra of connected bipartite 4-regular integral graphs. In Section 3 we give some upper bounds on the number of vertices in
r-regular bipartite graphs. From the results in this section and in [19] follows that,
except for 5 exceptional spectra, 4-regular bipartite integral graph has at most 1260
vertices. In Section 4 we give the list of known 4-regular integral graphs. Section 5
contains concluding remarks.
0 1991
Mathematics Subject Classification: 05C50
109
110
2
D. Cvetković, S. Simić, D. Stevanović
Spectra of 4-regular bipartite integral graphs
Suppose that G is a 4-regular bipartite integral graph. Regular bipartite graphs
have the same number of vertices in each part so that we may assume that they
have p = 2n vertices. Using superscripts to represent multiplicities, we shall write
its spectrum in the form
[4, 3x , 2y , 1z , 02w , −1z , −2y , −3x , −4].
Let further q and h denote the numbers of quadrilaterals and hexagons in G. It
is well known that the sum of k th powers of the eigenvalues is just the number
of closed walks of length k. On the other hand, these results provide us with the
following lemma:
Lemma 1 The parameters n, x, y, z, w, q, h satisfy the Diophantine equations:
1X 0
(1)
λi = 1 + x + y + z + w = n,
2
1X 2
(2)
λi = 16 + 9x + 4y + z = 4n,
2
1X 4
(3)
λi = 256 + 81x + 16y + z = 28n + 4q,
2
1X 6
(4)
λi = 4096 + 729x + 64y + z = 232n + 72q + 6h.
2
Another useful lemma is due to A. J. Hoffman [13]. Let a regular graph G
have distinct eigenvalues µ1 = r, µ2 , . . . , µk , and let A be the adjacency matrix of
G while J is the matrix all of whose entries are 1 and I is the identity matrix.
Lemma 2 The adjacency matrix A satisfies the equation
k
Y
(5)
(r − µi )J = p
i=2
k
Y
(A − µi I).
i=2
The search for possible spectra is divided in cases depending on the greatest
integer less than 4 avoided in the spectrum of G (see 1–5).
1
Case x = 0
By putting x = 0 in the equations (1)-(4) and then eliminating n and y from (1) and
(2) we get 3z + 4w = 12, and since z, w ≥ 0, the only possibilities are (z, w) = (0, 3)
or (z, w) = (4, 0).
Subcase (z, w) = (0, 3)
By substituting values of x, z and w in (1)-(4) and putting y = n − 4 from (1) into
(3) and (4) we get
(6)
3n + q
= 48,
4-Regular integral graphs
(7)
28n + 12q + h
111
= 640.
From (6) follows 3 | q, so q = 3q1 , and it reduces to n + q1 = 16, which shows that
n ≤ 16. Further, from (6) and (7) follows that h = 8n + 64. If y = 0 then n = 4 and
the corresponding spectrum is shown in Table 1. If y > 0 Hoffman’s identity (5)
reads 384J = 2n(A4 − 4A2 ) + 8n(A3 − 4A). Graph G is bipartite, and if we take
vertices u, v from distinct parts of it then (A2k )u,v = 0. Taking (u, v)−entries in the
Hoffman’s identity, we have 384 = 8n((A3 )u,v − 4Au,v ), wherefrom 8n | 384, i.e.
n | 48. Since n ≤ 16, the only possibilities for n are 6, 8, 12, 16. The corresponding
spectra are shown in Table 1. Here and in other tables of potential spectra the
column “Label” indicates the existence of graphs with the corresponding spectrum
by refering to the list of known 4-regular integral graphs in Section 4. If the graph
with the corresponding spectrum does not exist, then this column contains “—”. If
all the graphs with the given spectrum are known, then the column “All” contains
symbol “+”.
n
4
6
8
12
16
x
0
0
0
0
0
y
0
2
4
8
12
z
0
0
0
0
0
w
3
3
3
3
3
q
36
30
24
12
0
h
96
112
128
160
192
Label
I8,1
I12,4
I16,1−2
I24,1−2
I32,1
All
+
+
+
+
+
Table 1: Possible integral graph spectra with (x, z, w) = (0, 0, 3).
Subcase (z, w) = (4, 0)
By substituting values of x, z and w in (1)-(4) and putting y = n − 5 from (1) into
(3) and (4) we get
(8)
3n + q
(9)
28n + 12q + h
= 45,
= 630.
From (8) follows 3 | q, so q = 3q1 , and it reduces to n + q1 = 15, which shows that
n ≤ 15. Further, from (8) and (9) follows that h = 8n + 90. If y = 0 then n = 5
and the corresponding spectrum is shown in Table 2. If y > 0 Hoffman’s identity
(5) reads 1440J = 2n(A5 − 5A3 + 4A) + 8n(A4 − 5A2 + 4I), therefore n | 180, and
since n ≤ 15, the only possible values for n are 6, 9, 10, 12, 15. The corresponding
spectra are shown in Table 2.
112
D. Cvetković, S. Simić, D. Stevanović
n
5
6
9
10
12
15
x
0
0
0
0
0
0
y
0
1
4
5
7
10
z
4
4
4
4
4
4
w
0
0
0
0
0
0
q
30
27
18
15
9
0
h
130
138
162
170
186
210
Label
I10,1
I12,2
I18,1−3
I20,1−3
—
I30,1
All
+
+
+
+
+
Table 2: Possible integral graph spectra with (x, z, w) = (0, 4, 0).
2
Case x > 0, y = 0
By putting y = 0 in the equations (1)-(4) and subtracting (2) from (3) we obtain
240 + 72x = 24n + 4q, from which q = 6q 0 , and
3x = n + q 0 − 10.
(10)
On the other hand, subtracting (1) from (2) yields
(11)
8x = 3n + w − 15.
Eliminating x from (10) and (11) we obtain n = 8q 0 − 3w − 35. From the fact
that q 0 , w, x, z, h are non-negative integers, we get the following system of linear
inequalities:
x=
3q 0 − w − 15
z = 5q 0 − 3w − 21
h = 210 − 16q 0 − 6w
(12)
(13)
(14)
≥ 0,
≥ 0,
≥ 0.
Hoffman’s identity (5) in this case reads 3360J = 2n(A6 − 10A4 + 9A2 ) + 8n(A5 −
10A3 + 9A). Thus, we have n | 420. Solving the system (12) - (14) for q 0 and w,
and eliminating the spectra not satisfying n | 420, we get all the possibilities shown
in Table 3. Nonexistence of graphs with spectra indicated by “—” in this table is
shown in [19].
3
Case x > 0, y > 0, z = 0
By putting z = 0 in the equations (1)-(4) and eliminating n from (1) and (2) we
obtain 4w = 12 + 5x, so that x = 4x1 . Substituting the value of n from (2) into (3)
we get q = 36 + 18x1 − 3y, so that q = 3q1 and
(15)
1 This
q1 = 12 + 6x1 − y.
will be proven elsewhere.
4-Regular integral graphs
n
7
10
12
14
15
20
21
28
42
x
1
2
3
4
4
6
6
9
14
y
0
0
0
0
0
0
0
0
0
z
3
6
5
4
8
10
14
15
26
w
2
1
3
5
2
3
0
3
1
q
36
36
42
48
42
48
42
54
60
h
102
108
80
52
86
64
98
84
44
113
Label All
I14,1 +1
—
—
—
Table 3: Possible integral graph spectra with x > 0, y = 0.
Now substituting the value of n from (2) and the value of q1 from (15) into (4) we
have 21 h = 48 − 39x1 + 4y ≥ 0, therefore
(16)
y ≥ 9.75x1 − 12.
From (15) and (16) follows
(17)
0 ≤ q1 ≤ 24 − 3.75x1 ,
which gives that 1 ≤ x1 ≤ 6. Eliminating y from (2) and (15) we get n = 15x1 −
q1 + 16, and from (17) follows 18.75x1 − 8 ≤ n ≤ 15x1 + 16. Hoffman’s identity
(5) now reads
2688J = 2n(A6 − 13A4 + 36A2 ) + 8n(A5 − 13A3 + 36A),
and therefore n | 336. All the possible spectra in this case are shown in Table 4.
n
14
16
21
24
28
42
56
x
4
4
4
4
4
8
12
y
1
3
8
11
15
20
25
z
0
0
0
0
0
0
0
w
8
8
8
8
8
13
18
q
51
45
30
21
9
12
15
h Label
26
42
82
106
138
100
62
Table 4: Possible integral graph spectra with x, y > 0, z = 0.
114
4
D. Cvetković, S. Simić, D. Stevanović
Case x > 0, y > 0, z > 0, w = 0
Eliminating y and z from the equations (1) and (2) we get y = n − 38 x − 5 and
z = 53 x + 4, from which x = 3x1 (x1 ∈ N). Eliminating then q and h from the
equations (3) and (4), we obtain q = −3n + 30x1 + 45 and h = 8n − 80x1 + 90.
From the inequalities y ≥ 1, q ≥ 0, h ≥ 0, we get the condition
max {1,
n
14
15
18
20
21
28
30
30
35
35
36
42
42
45
45
60
60
63
63
63
70
70
70
84
84
84
90
90
90
x
3
3
3
3
3
6
6
9
6
9
9
9
12
9
12
15
18
15
18
21
18
21
24
21
24
27
24
27
30
y
1
2
5
7
8
7
9
1
14
6
7
13
5
16
8
15
7
18
10
2
17
9
1
23
15
7
21
13
5
z
9
9
9
9
9
14
14
19
14
19
19
19
24
19
24
29
34
29
34
39
34
39
44
39
44
49
44
49
54
w
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
q
33
30
21
15
12
21
15
45
0
30
27
9
39
0
30
15
45
6
36
66
15
45
75
3
33
63
15
45
75
n
3
n
9 n 3
− } ≤ x1 ≤ min { + , − }.
10 2
10 8 8
4
h
122
130
154
170
178
154
170
90
210
130
138
186
106
210
130
170
90
194
114
34
170
90
10
202
122
42
170
90
10
Label
—
—
I60,3
I70,1
—
I90,1
—
—
—
—
—
n
105
105
105
126
126
140
140
140
180
180
180
210
210
210
252
252
252
315
315
315
420
420
420
630
630
630
1260
1260
1260
x
27
30
33
36
39
39
42
45
51
54
57
60
63
66
72
75
78
90
93
96
123
126
129
186
189
192
375
378
381
y
28
20
12
25
17
31
23
15
39
31
23
45
37
29
55
47
39
70
62
54
87
79
71
129
121
113
255
247
239
z
49
54
59
64
69
69
74
79
89
94
99
104
109
114
124
129
134
154
159
164
209
214
219
314
319
324
629
634
639
w
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
q
0
30
60
27
57
15
45
75
15
45
75
15
45
75
9
39
69
0
30
60
15
45
75
15
45
75
15
45
75
h
210
130
50
138
58
170
90
10
170
90
10
170
90
10
186
106
26
210
130
50
170
90
10
170
90
10
170
90
10
Label
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
Table 5: Possible integral graph spectra with x, y, z > 0, w = 0.
4-Regular integral graphs
115
Hoffman’s identity (5) in this case is reduced to
10080 · J = 2n(A7 − 14A5 + 49A3 − 36A) + 8n(A6 − 14A4 + 49A2 − 36I),
therefore n | 1260, and we can easily find all possible spectra in this case. These
are shown in Table 5. Nonexistence of graphs with spectra indicated by “—” in
this table is shown in [19].
5
Case x > 0, y > 0, z > 0, w > 0
Eliminating z, w, q, h from the equations (1)-(4) of Lemma 1, we get
z
w
q
=
=
=
−9x − 4y + 4n − 16,
8x + 3y − 3n + 15,
18x + 3y − 6n + 60,
h
=
−96x − 26y + 34n − 40.
Hoffman’s identity (5) in this case is reduced to
40320 · J = 2n(A8 − 14A6 + 49A4 − 36A2 ) + 8n(A7 − 14A5 + 49A3 − 36A),
therefore n | 5040. From this and the inequalities z ≥ 1, w ≥ 1, q ≥ 0, h ≥ 0, we
can easily (using a computer program) obtain all possible spectra. There are 1803
such spectra. In Subsection 3 we show that the upper bound for n is 3280, hence
the case n = 5040 is impossible. In [19] graph angles are exploited to show the
nonexistence of graphs with some of these spectra, reducing the total number of
potential spectra in this case to 1259. It is also shown there that there are only
5 potential spectra with 630 < n ≤ 2520. In Table 7 we have shown those with
n ≤ 20 and these 5 exceptional spectra. In Table 6 for each n | 5040, n ≤ 630 we
give the number SP of potential spectra.
n
21
24
28
30
35
36
SP
12
14
17
18
22
23
n
40
42
45
48
56
60
SP
24
27
29
31
35
33
n
63
70
72
80
84
SP
38
40
35
35
39
n
90
105
112
120
126
SP
44
44
39
44
47
n
140
144
168
180
210
SP
40
42
32
37
47
n
240
252
280
315
336
SP
28
32
40
47
14
n
360
420
504
560
630
SP
17
22
33
40
47
Table 6: Number of potential integral graph spectra with x, y, z, w > 0 and 21 ≤
n ≤ 630.
116
D. Cvetković, S. Simić, D. Stevanović
n
8
9
10
10
12
12
12
12
14
14
14
14
14
15
15
15
15
15
15
16
16
16
16
16
16
16
x
1
1
1
2
1
2
2
3
1
2
2
3
3
1
2
2
3
3
4
1
2
2
3
3
4
4
y
1
2
3
1
5
2
3
1
7
4
5
2
3
8
5
6
3
4
1
9
6
7
4
5
1
2
z
3
3
3
2
3
6
2
1
3
6
2
5
1
3
6
2
5
1
4
3
6
2
5
1
8
4
w
2
2
2
4
2
1
4
6
2
1
4
3
6
2
1
4
3
6
5
2
1
4
3
6
2
5
q
33
30
27
39
21
30
33
45
15
24
27
36
39
12
21
24
33
36
45
9
18
21
30
33
39
42
h
110
118
126
82
142
124
98
54
158
140
114
96
70
166
148
122
104
78
60
174
156
130
112
86
94
68
Label
I20,5
I24,3
I30,2−3
n
18
18
18
18
18
18
18
18
18
20
20
20
20
20
20
20
20
20
20
20
720
840
1260
1680
2520
x
1
2
2
3
3
4
4
5
5
2
2
3
3
4
4
5
5
5
6
6
208
244
370
496
748
y
11
8
9
6
7
3
4
1
2
10
11
8
9
5
6
2
3
4
1
2
172
196
280
364
532
z
3
6
2
5
1
8
4
7
3
6
2
5
1
8
4
11
7
3
6
2
304
364
574
784
1204
w
2
1
4
3
6
2
5
4
7
1
4
3
6
2
5
1
4
7
6
9
35
35
35
35
35
q
3
12
15
24
27
33
36
45
48
6
9
18
21
27
30
36
39
42
51
54
0
0
0
0
0
h
190
172
146
128
102
110
84
66
40
188
162
144
118
126
100
108
82
56
38
12
0
0
0
0
0
Label
I36,1
I36,2
I36,3
I40,1−2
Table 7: Possible integral graph spectra with x, y, z, w > 0 and n ≤ 20, and 5 exceptional spectra.
3
Upper bound on the number of vertices
Here we give the upper bound on the number of vertices in regular bipartite graph,
from which follows that there are no 4-regular bipartite integral graphs with n =
5040. This upper bound presents improvement in case of bipartite graphs upon
previously known upper bound (see [11]) for regular graphs of given diameter.
Theorem 3 Let p be the number of vertices in connected r-regular bipartite graph
G = (V, E) with radius R. Then
p≤
2(r − 1)R − 2
.
r−2
4-Regular integral graphs
117
Proof Let u be a vertex of G with eccentricity R. Let Nk (u) = {v ∈ V |d(u, v) = k}
be the k th neighborhood of u, and dk (u) = |Nk (u)| the number of vertices in it.
Then d0 (u) = 1 and d1 (u) = r. When 2 ≤ k ≤ R−1, it holds dk (u) ≤ (r−1)dk−1 (u),
since each vertex of Nk−1 (u) may be adjacent to at most r − 1 vertices of Nk (u).
Therefrom by induction dk (u) ≤ r(r − 1)k−1 for 2 ≤ k ≤ R − 1. On the other hand,
each vertex of NR (u) is adjacent to r vertices of NR−1 (u) (since G is bipartite no
R−1
two vertices of NR (u) may be adjacent) so dR (u) ≤ r−1
. Now
r dR−1 (u) ≤ (r − 1)
R
PR
2(r−1) −2
p = k=0 dk (u) ≤
, and the theorem is proved. 2
r−2
By a theorem from [5], for the diameter D of a connected graph G holds
D ≤ s − 1, where s is the number of distinct eigenvalues of G. When G is connected
regular integral graph of degree r, we get D ≤ 2r. From Theorem 3 then follows
that connected bipartite 4-regular integral graph has diameter at most 8 and it
may have at most p = 2n ≤ 6560 vertices. But from the previous section it can
be seen that the highest possible value for n not greater than 3280 is 2520. This
bound cannot be lowered, but using graph angles it is shown in [19] that there are
only 5 possible spectra with 630 < n ≤ 2520.
4
A list of known 4-regular integral graphs
In this section we give a list of 65 connected 4-regular integral graphs. A part of
these graphs are known in the literature. The others appear here for the first time
and are derived from the known graphs by some construction procedures described
below. We give to each of these graphs a label of the form Ip,a where p is the
number of vertices, and a is the ordering number of graph in the group of graphs
with the same number of vertices. For each graph we give its spectrum and a short
description of the graph, sometimes with references to the literature. Graphs in
the list are sorted by the number of vertices and then by the spectra.
Operations + and × denote the usual graph sum and product. Operation ⊕
denotes the strong sum: given two graphs G and H, with vertex sets V (G) and
V (H), their strong sum G ⊕ H is the graph with vertex set V (G) × V (H) in which
two vertices (x, a) and (y, b) are adjacent if and only if a is adjacent to b in H and
either x is adjacent to y in G or x = y. L(G) is the line graph of G, S(G) the
subdivision graph, and finally L2 (G) = L(S(G)).
Graphs G1 , . . . , G13 are connected cubic integral graphs from [15]. Graphs
G1 , . . . , G8 are bipartite, while G9 , . . . , G13 are not. These graphs are also found
in [6, 3], but in different order.
In the following list we give their spectra.
G1 = K3,3 : [3, 04 , −3].
G2 = K2 + K2 + K2 = G9 × K2 (the cube graph): [3, 13 , −13 , −3].
G3 (Tutte’s 8-cage): [3, 29 , 010 , −29 , −3].
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D. Cvetković, S. Simić, D. Stevanović
G4 = G10 × K2 = G11 × K2 (the Desargues graph): [3, 24 , 15 , −15 , −24 , −3].
G5 : [3, 24 , 15 , −15 , −24 , −3].
G6 : [3, 2, 12 , 02 , −12 , −2, −3].
G7 = K2 + C6 = G12 × K2 (the 6-sided prism): [3, 22 , 1, 04 , −1, −22 , −3].
G8 = G13 × K2 : [3, 26 , 13 , 04 , −13 , −26 , −3].
G9 = K4 : [3, −13 ].
G10 (the Petersen graph): [3, 15 , −24 ].
G11 : [3, 2, 13 , −12 , −23 ].
G12 = K2 + K3 (the 3-sided prism): [3, 1, 02 , −22 ].
G13 = L2 (K4 ) : [3, 23 , 02 , −13 , −23 ].
For each Gi (i = 1, . . . , 13) the graphs L(Gi ), L(Gi )×K2 , Gi +K2 and Gi ⊕K2
are connected 4-regular integral graphs. In the Table 8 we give their positions in
the list of 4-regular integral graphs. If Gi is bipartite, then Gi + K2 = Gi ⊕ K2 , so
we do not double the column for these graphs.
i
1
2
3
4
5
6
7
8
L(Gi ) L(Gi ) × K2 Gi + K2
I9,2
I18,1
I12,2
I12,7
I24,2
I16,1
I45,1
I90,1
I60,3
I30,4
I60,1
I40,1
I30,5
I60,2
I40,2
I15,3
I30,2
I20,5
I18,5
I36,1
I24,3
I36,4
I72,1
I48,1
i
9
10
11
12
13
L(Gi ) L(Gi ) × K2 Gi + K2 Gi ⊕ K2
I6,1
I12,4
I8,2
I8,1
I15,2
I30,1
I20,4
I20,1
I15,4
I30,3
I20,7
I20,6
I9,4
I18,1
I12,5
I12,2
I18,6
I36,2
I24,5
I24,4
Table 8: Positions of graphs L(Gi ), L(Gi ) × K2 , Gi + K2 and Gi ⊕ K2 in the list.
Let Gi be nonbipartite, i.e. i = 9, 10, . . . , 13. Graphs (Gi + K2 ) × K2 and
(Gi × K2 ) + K2 are connected, 4-regular integral graphs which, in addition, are
cospectral. However, we have the following proposition.
Proposition 1 Let G be a nonbipartite graph. Graphs (G + K2 ) × K2 and (G ×
K2 ) + K2 are isomorphic.
Proof For vertices of (G + K2 ) × K2 and (G × K2 ) + K2 we will shortly write uab
instead of (u, a, b). Then (u0 a0 b0 , u1 a1 b1 ) ∈ E((G + K2 ) × K2 ) ⇔ (u0 a0 , u1 a1 ) ∈
E(G + K2 ) and b0 6= b1 ⇔ (u0 , u1 ) ∈ E(G), a0 = a1 , b0 6= b1 or u0 = u1 , a0 6=
4-Regular integral graphs
119
a1 , b0 6= b1 ; and (u0 a0 b0 , u1 a1 b1 ) ∈ E((G × K2 ) + K2 ) ⇔ (u0 a0 , u1 a1 ) ∈ E(G ×
K2 ), b0 = b1 or u0 a0 = u1 a1 , b0 6= b1 ⇔ (u0 , u1 ) ∈ E(G), a0 6= a1 , b0 = b1 or
u0 = u1 , a0 = a1 , b0 6= b1 . Now it is not hard to see that the mapping f : V ((G +
K2 ) × K2 ) 7→ V ((G × K2 ) + K2 ) given by f (u, a, b) = (u, a + b, a), where addition
is modulo 2, is actually the isomorphism of these graphs. 2
Hence, all graphs (Gi + K2 ) × K2 and (Gi × K2 ) + K2 for i = 9, 10, . . . , 13
are contained in the last row of Table 8.
Graphs D1 , . . . , D16 and E1 , . . . , E8 are connected 4-regular integral graphs
as found in [17]. Graphs D1 , . . . , D16 appear in the list as graphs labelled by I8,1 ,
I10,1 , I12,4 , I12,2 , I16,1 , I16,2 , I24,1 , I24,2 , I32,1 , I18,1 , I18,2 , I18,3 , I20,1 , I20,2 , I20,3 ,
I30,1 , while E1 , . . . , E8 are labelled by I5,1 , I6,1 , I8,2 , I12,6 , I12,7 , I9,4 , I9,2 , I15,2 .
All graphs in the list with at most 12 vertices have also been found in the
technical report [1] by means of computer search, and they are displayed in Figure 21 of [1]. Note that in the graph no. 17 from that figure, which is isomorphic
to E5 , two edges are misplaced and compare it with the drawing given in [17].
5
Concluding remarks
Potential spectra of connected 4-regular bipartite integral graphs, determined in
Section 2, are quite numerous and we cannot expect that all 4-regular integral
graphs will be determined in near future. Nonexistence results for many of these
potential spectra will be obtained by considering appropriate graph properties but
many others will remain to be considered.
Potential spectra from subsection 1 (case x = 0), i.e. those from Tables 1
and 2, have been considered in [17] and all the corresponding graphs determined.
They are graphs D1 , . . . , D16 . Other known 4-regular integral graphs are presented
in Section 4.
120
D. Cvetković, S. Simić, D. Stevanović
I12,6 = E4 : [4, 23 , 03 , −25 ], c.f. graph
no. 9 in Table 9.1 in [4].
The list:
I5,1 = E1 = K5 : [4, −14 ].
I6,1 = E2 = L(K4 ) = L(G9 ) : [4, 03 , −22 ],
also NEPS of C3 and K2 with basis
{(1, 1), (1, 0)}, c.f. [18].
I7,1 = C4 ∪ C3 : [4, 1, 02 , −12 , −3], c.f.
graph 7402 in tables of [10].
6
I8,1 = D1 = G9 ⊕ K2 = K4,4 : [4, 0 , −4],
c.f. graph 8406 in tables of [10].
3
I8,2 = E3 = G9 + K2 = L(K2,4 ) : [4, 2, 0 ,
−23 ], c.f. graph 8401 in tables of [10].
I9,1 : [4, 13 , 02 , −22 , −3], c.f. graph 9414
in tables of [10].
I9,2=E7=L(G1 )=C3 +C3=C3 ×C3 : [4,14 ,
−24 ], c.f. graph 9410 in tables of [10].
I9,3 : [4, 2, 1, 02 , −12 , −2, −3], c.f. graph
9407 in tables of [10].
I9,4 = E6 = L(G12 ) : [4, 2, 12 , −12 , −23 ],
c.f. graph 9404 in tables of [10].
I10,1=I5,1×K2=D2=5K2 : [4,14 ,−14 ,−4].
6
2
2
I12,7 = E5 = L(G2 ) : [4, 23 , 03 , −25 ].
I12,8 : [4,3,13 ,0,−13 ,−22 ,−3], c.f. graph
no. 18 in Figure 21 of [1].
I14,1 = I7,1 ×K2 : [4,3,13 ,04 ,−13 ,−3,−4].
I15,1 : [4, 24 , 1, 02 , −12 , −24 , −3], c.f.
graph X from [17].
I15,2 = E8 = L(G10 ) : [4, 25 , −14 , −25 ].
I15,3 = L(G6 ) : [4, 3, 22 , 12 , 02 , −1, −26 ].
I15,4 = L(G11 ) : [4, 3, 23 , 02 , −13 , −25 ].
I16,1 = I8,2 × K2 = D5 = G2 + K2 =
C4 + C4 : [4, 24 , 06 , −24 , −4].
I16,2 = D6 : [4, 24 , 06 , −24 , −4].
I18,1 = I9,2 × K2 = D10 = L(G1 ) × K2 =
L(G12 ) × K2 = (C3 + C3 ) × K2 : [4, 24 , 14 ,
−14 , −24 , −4].
I18,2 = D11 : [4, 24 , 14 , −14 , −24 , −4].
I18,3 = D12 : [4, 24 , 14 , −14 , −24 , −4].
I12,1 : [4, 1 , 0, −2 , −3 ], NEPS of C3 ,
K2 and K2 with basis {(1, 1, 1), (0, 0, 1),
(0, 1, 0)}, c.f. [18].
I18,4 = C3 + C6 : [4, 32 , 14 , 05 , −24 , −32 ].
I12,2 = D4 = G1 + K2 = G12 ⊕ K2 :
[4, 2, 14 , −14 , −2, −4], also NEPS of C3 ,
K2 and K2 with basis {(1, 1, 1), (0, 1, 1),
(0, 0, 1)}, c.f. [18].
I18,6 = L(G13 ) : [4, 33 , 12 , 03 , −13 , −26 ].
I12,3 : [4,2,14 ,0,−12 ,−2,−32 ], c.f. graph
no. 12 in Figure 21 of [1].
I20,2 = D14 : [4, 25 , 14 , −14 , −25 , −4].
I18,5 = L(G7 ) : [4, 32 , 2, 14 , 0, −12 , −27 ].
I20,1 = D13 = G10 ⊕ K2 : [4, 25 , 14 , −14 ,
−25 , −4].
I20,3 = D15 : [4, 25 , 14 , −14 , −25 , −4].
I12,4 = I6,1 × K2 = D3 = L(G9 ) × K2 =
C3 × C4 : [4, 22 , 06 , −22 , −4].
I20,4 = G10 + K2 : [4, 26 , 05 , −14 , −34 ].
I12,5 = G12 + K2 = C3 + C4 : [4, 22 , 12 , 0,
−14 , −32 ].
I20,5 = G6 + K2 : [4, 3, 23 ,13 ,04 ,−13 ,−23 ,
−3, −4].
4-Regular integral graphs
121
I20,6 = G11 ⊕ K2 : [4, 3, 23 , 13 , 04 , −13 ,
−23 , −3, −4].
I36,3 = C6 + C6 = (C3 + C6 ) × K2 :
[4, 34 , 24 , 14 , 010 , −14 , −24 , −34 , −4].
I20,7 = G11 + K2 : [4, 3, 24 , 1, 05 , −13 ,
−22 , −33 ].
I36,4=L(G8 ) : [4,36 ,23 ,14 ,03 ,−16 ,−213 ].
4
5
4
6
I20,8 = L2 (K5 ) : [4, 3 , 0 , −1 , −2 ]. It is
proved in [7] that L2 (G) (G connected
with at least two vertices) is integral if
and only if G is complete graph.
I24,1 = D7 : [4, 28 , 06 , −28 , −4].
I24,2=D8=L(G2 )×K2 : [4,28 ,06 ,−28 ,−4].
I24,3 = G7 + K2 = C4 + C6 : [4, 32 , 22 , 16 ,
02 , −16 , −22 , −32 , −4].
I24,4 = G13 ⊕ K2 : [4, 33 , 15 , 06 , −15 ,
−33 , −4].
I24,5 = G13 + K2 : [4, 33 , 2, 15 , 03 , −15 ,
−23 , −33 ].
10
4
I30,1 = D16 = L(G10 ) × K2 : [4, 2 , 1 ,
−14 , −210 , −4].
I30,2 = L(G6 )×K2 = I15,1 ×K2 : [4, 3, 28 ,
13 , 04 , −13 , −28 , −3, −4].
I30,3 = L(G11 )×K2 : [4, 3, 28 , 13 , 04 , −13 ,
−28 , −3, −4].
I30,4 = L(G4 ) : [4, 34 , 25 , 05 , −14 , −211 ].
I30,5 = L(G5 ) : [4, 34 , 25 , 05 , −14 , −211 ].
I32,1 = D9 : [4, 212 , 06 , −212 , −4].
I35,1 : [4, 214 , −114 , −36 ], the odd graph
O4 [14].
I36,1 = L(G7 )×K2 : [4, 32 , 28 , 16 , 02 , −16 ,
−28 , −32 , −4].
I36,2 = L(G13 ) × K2 : [4, 33 , 26 , 15 , 06 ,
−15 , −26 , −33 , −4].
I40,1 = G4 + K2 : [4, 34 , 26 , 14 , 010 , −14 ,
−26 , −34 , −4].
I40,2 = G5 + K2 = (G10 + K2 ) × K2 =
(G11 + K2 ) × K2 = L2 (K5 ) × K2 : [4, 34 ,
26 , 14 , 010 , −14 , −26 , −34 , −4].
It is interesting to note that the graphs
G10 +K2 , G11 +K2 and L2 (K5 ) all have
mutually different spectra.
I45,1 = L(G3 ) : [4, 39 , 110 , −19 , −216 ].
I48,1 = G8 + K2 : [4, 36 , 24 , 110 , 06 , −110 ,
−24 , −36 , −4].
I60,1 = L(G4 ) × K2 : [4, 34 , 216 , 14 , 010 ,
−14 , −216 , −34 , −4].
I60,2 = L(G5 ) × K2 : [4, 34 , 216 , 14 , 010 ,
−14 , −216 , −34 , −4].
I60,3 = G3 + K2 : [4, 39 , 2, 119 , −119 , −2,
−39 , −4].
I70,1 = I35,1 × K2 : [4, 36 , 214 , 114 , −114 ,
−214 , −36 , −4].
I72,1 = L(G8 ) × K2 : [4, 36 , 216 , 110 , 06 ,
−110 , −216 , −36 , −4].
I90,1 = L(G3 ) × K2 : [4, 39 , 216 , 119 , −119 ,
−216 , −39 , −4].
122
D. Cvetković, S. Simić, D. Stevanović
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[email protected]
[email protected]
Faculty of Electrical Engineering,
University of Belgrade,
P.O. Box 35-54, 11120 Belgrade, Yugoslavia
[email protected]
Department of Mathematics,
Faculty of Philosophy,
University of Niš,
Ćirila i Metodija 2, 18000 Niš, Yugoslavia
(Received October 7, 1998)