# Small integral trees

An integral graph is a graph with integral eigenvalues. A tree is a connected graph without cycles.

There are 0, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320, 48629, 123867, 317955, 823065, 2144505, 5623756, 14828074, 39299897, 104636890, 279793450, 751065460, 2023443032, 5469566585, 14830871802, 40330829030, 109972410221, 300628862480, 823779631721, 2262366343746, 6226306037178, 17169677490714, 47436313524262, 131290543779126, 363990257783343 trees on 0–40 vertices, respectively. This is Sequence A000055 in Sloane's On-Line Encyclopedia of Integer Sequences. (Except that he starts with 1, but there are no trees on 0 vertices: just like 1 is not a prime number but a product of zero primes, the empty graph is not connected, but a forest with zero trees.)

Only very few of all these trees have only integral eigenvalues. Examples are known of diameters 0–8 and 10. It  is  was unknown whether integral trees of arbitrary diameter exist. (Péter Csikvári (2010) has constructed integral trees with arbitrary even diameter, see below. Ebrahim Ghorbani et al. (2011) settled the odd-diameter case.)

There are many papers on integral trees, but I have not seen tables with a complete enumeration of all small integral trees. Here a table with all on at most 50 vertices.

## The integral trees on at most 50 vertices

# is a serial number, n is the number of vertices, d the diameter.

Since the spectrum is symmetric around 0, it suffices to give the nonnegative half. Multiplicities are written as exponents.

The tree is given by a level sequence.

 # n d name tree spectrum 1 1 0 K1 0 0 2 2 1 K2 01 1 3 5 2 K1,4 01111 2,03 4 6 3 K1,2~K1,2 012211 2,1,02 5 7 4 SK1,3 0121212 2,12,0 6 10 2 K1,9 0111111111 3,08 7 14 3 K1,6~K1,6 01222222111111 3,2,010 8 17 2 K1,16 01111111111111111 4,015 9 17 4 SK1,8 01212121212121212 3,17,0 10 17 4 K1,7~SK1,4 01222222212121212 3,2,13,07 11 19 4 K1,5~SK1,6 0122222121212121212 3,2,15,05 12 25 5 T(2)*T(3,4)~T(3,1) 0123333233332333322121212 3,23,13,011 13 26 2 K1,25 01111111111111111111111111 5,024 14 26 4 T(5,4) 01222212222122221222212222 3,24,016 15 26 3 K1,12~K1,12 01222222222222111111111111 4,3,022 16 31 4 SK1,15 0121212121212121212121212121212 4,114,0 17 31 6 T(1)*T(2,4)*T(1,1,3)*T(2,3,1) 0123331232323123232312222122221 3,24,15,011 18 31 6 T(3,4)*T(3,1,3) 0123331233312333122221222212222 3,25,1,017 19 35 4 K1,13~SK1,10 01222222222222212121212121212121212 4,3,19,013 20 37 2 K1,36 0111111111111111111111111111111111111 6,035 21 37 4 K1,11~SK1,12 0122222222222121212121212121212121212 4,3,111,011 22 37 6 T(1,5,3,1) 0123232312323231232323123232312323231 3,24,111,05 23 37 6 T(1,4)*T(2,1,3)*T(3,3,1) 0123331233312323231232323123232312222 3,25,17,011 24 42 3 K1,20~K1,20 01^{20}12^{20} 5,4,038 25 46 4 T(6,4)*T(1,14) 012^{14}(12222)^6 4,3,25,032 26 49 4 SK1,24 0(12)^{24} 5,123,0 27 50 2 K1,49 01^{49} 7,048 28 50 4 T(4)*T(9,4) 0(12222)^91^4 4,28,1,030

Symbols used in the graph name:

Km is the complete graph on m vertices.

K1,m is the star on m+1 vertices. (It has an obvious center.)

SX is the subdivision of X. (It has the same center as X, if any.)

G~H is the result of joining the center of G with the center of H by an edge. (Here `center' is defined by induction. The result does not have a center.)

C(G1,...) is the cone over G1,..., that is, the result of joining a new vertex to the center of each of the Gi. (The new vertex is the center.) For example, K1,m is C(mK1), and SK1,m is C(mK2).

T(nk,...,n1) is defined by induction on k as the graph C(nkT(nk-1,...,n1)), where T() is a single vertex. For example, K1,m is T(m), and SK1,m is T(m,1).

G*H is the graph obtained from the disjoint union of G and H by identifying the centers of G and H. For example, C(mK1,t+sK1) is T(s)*T(m,t).

## Spectra

### K1,m

The spectrum of K1,m is
√m, 0m–1, –√m.
It follows that K1,m is integral when m is a square (Harary & Schwenk (1974)). This covers #1, 2, 3, 6, 8, 13, 20, 27 (since #1 is K1,0 and #2 is K1,1). (Harary & Schwenk also gave the three examples #4, 5, 7.)

### SK1,m

The spectrum of the graph SK1,m is
√(m+1), 1m–1, 0, (–1)m–1, –√(m+1).
It follows that SK1,m is integral iff m+1 is a square. This covers #5, 9, 16, 26.

Watanabe & Schwenk (1979) showed that these graphs are the only integral trees with a single vertex of degree more than two.

They also studied the situation with two adjacent vertices of degree more than two, and proved that in that situation one has either K1,m~K1,r or K1,m~SK1,r. (They were unable to handle the case of two nonadjacent vertices of degree more than two, but using interlacing and Godsil's Lemma it is not difficult to see that that does not yield any further examples - see Brouwer (2009).)

### K1,m~K1,r

The former is integral iff
x4 – (m+r+1)x2 + mr
has only integral roots. (The spectrum consists of 0m+r–2 together with these four roots.) This covers #4, 7, 15, 24 (with m = r = 2, 6, 12, 20). The next smallest case is m = r = 30 on 62 vertices. Examples where this equation has integral roots only are the cases with m = r = a(a+1) for some integer a. Now the positive roots are a and a+1. But there are also other solutions - the smallest is K1,50~K1,98 on 150 vertices. The question which m and r give integral solutions was settled by Graham (1980).

### K1,m~SK1,r

The latter is integral iff
x4 – (m+r+2)x2 + mr + m + 1
has only integral roots. (The spectrum consists of (–1)r–1, 0m, 1r–1 together with these four roots.) This covers #10, 11, 19, 21 (with (m,r) = (7,4), (5,6), (13,10), (11,12)). The next smallest cases are (m,r) = (21,18), (19,20) on 59 and 61 vertices, respectively. The m and r that give integral solutions can be found by the method of Graham (1980). See also below.

### T(r,m)

They also studied the graphs T(r,m), and found that these are integral iff both m and r+m are squares. (The spectrum consists of –√(r+m), (–√m)r–1, 0mr–r+1, (√m)r–1, √(r+m).) This covers #14 (with (r,m) = (5,4)). The next smallest cases are T(12,4) on 61 vertices and T(7,9) on 71 vertices.

### T(s)*T(m,t)~T(q,r)

Wang (2005), Corollary 4.1.3, studied the graphs T(s)*T(m,t)~T(q,r) of diameter 5, and found that these are integral provided
x4 – (m+t+s+1)x2 + st + r
has only integral roots, and t is a square, and either q=1 or r is a square. (The positive spectrum consists of 0 with multiplicity m(t–1)+q(r–1)+s, and √t with multiplicity m, and √r with multiplicity q–1, and the two positive roots of the above polynomial.)

The graph T(2)*T(3,4)~T(3,1) on 25 vertices was given by Wang (2005), p. 58. This is #12. It is the smallest integral tree of diameter 5.

The next smallest case is T(7)*T(3,9)~T(8,1) on 55 vertices.

### The remaining cases

Graph #17 is perhaps given here for the first time. It is T(1)*T(2,4)*T(1,1,3)*T(2,3,1).

The graph T(3,4)*T(3,1,3) on 31 vertices was given by Ligong Wang (2005), p. 68. This is #18.

Graphs #17, 18 are the smallest integral trees of diameter 6.

The graph T(1,5,3,1) on 37 vertices is due to Yao (2001) (cf. Wang (2005), p. 47). This is #22.

Graph #23 is perhaps given here for the first time. It is T(1,4)*T(2,1,3)*T(3,3,1).

The graph T(6,4)*T(1,14) on 46 vertices may be due to Yuan (1998). This is #25.

The graph T(4)*T(9,4) on 50 vertices was given by Watanabe (1979). This is #28.

### T(a)*T(b,1)*T(c,4)*T(d,1,3)*T(e,3,1)

The tree 01^a(12)^b(12222)^c(12333)^d(1232323)^e will be integral when it is 01^a (that is K1,a) with a = t2, or 0(12)^b (that is, SK1,b) with b = t2–1, or 0(12222)^c (that is, T(c,4)) with c = t2–4, or when b=0, c=3a+2d–3, e=t2–4a–3d. In this last case the nonnegative eigenvalues are 0, 1, 2, t with multiplicities 10a+9d+e–10, 2e+1, 3a+3d+e–4, 1. For t=3 and (a,d)=(2,0), (1,1), (0,3), (1,0), (0,2) this yields examples #12, 17, 18, 22, 23.

## More small examples

Without claiming completeness, let us give here some further integral trees on at most 100 vertices. We condense level sequences by writing e.g. 01^{16} instead of 01111111111111111, and 0(12)^3 instead of 0121212. We try to give the original discoverer, but have no access to many of the Chinese publications about this topic. WS stands for Watanabe & Schwenk (1979).

 n d name tree spectrum reference 55 5 T(7)*T(3,9)~T(8,1) 01(23^9)^32^7(12)^8 4,33,2,17,031 Wang (2005), p. 57 56 6 T(3)*T(8,4)*T(1,1,3)*T(1,3,1) 0111(12222)^8123331232323 4,29,13,030 new 59 4 K1,21~SK1,18 012^{21}(12)^{18} 5,4,117,021 WS 61 4 K1,19~SK1,20 012^{19}(12)^{20} 5,4,119,019 WS 61 4 T(12,4) 0(12222)^{12} 4,211,037 WS 62 3 K1,30~K1,30 012^{30}1^{30} 6,5,058 WS 62 4 T(10,4)*T(1,10) 012^{10}(12222)^{10} 4,3,29,040 Yuan (1998)? 62 6 T(3)*T(6,4)*T(4,3,1) 0111(12222)^6(1232323)^4 4,29,19,024 Wang (2005), p. 76 62 6 T(2)*T(7,4)*T(2,1,3)*T(2,3,1) 011(12222)^7(12333)^2(1232323)^2 4,210,15,030 new 62 6 T(1)*T(8,4)*T(4,1,3) 01(12222)^8(12333)^4 4,211,1,036 Wang (2005), p. 77 65 2 K1,64 01^{64} 8,063 WS 68 6 T(2)*T(5,4)*T(1,1,3)*T(5,3,1) 011(12222)^5(12333)(1232323)^5 4,210,111,024 new 68 6 T(1)*T(6,4)*T(3,1,3)*T(3,3,1) 01(12222)^6(12333)^3(1232323)^3 4,211,17,030 new 68 6 T(7,4)*T(5,1,3)*T(1,3,1) 0(12222)^7(12333)^5(1232323) 4,212,13,036 new 71 4 SK1,35 0(12)^{35} 6,134,0 WS 71 4 T(7,9) 0(12^9)^7 4,36,057 WS 71 6 T(6)*T(2,9)*T(2,8,1)*T(1,1,8) 01^6(12^9)^2(1(23)^8)^2123^8 4,34,2,114,031 new 74 6 T(2)*T(3,4)*T(8,3,1) 011(12222)^3(1232323)^8 4,210,117,018 Wang (2005), p. 76 74 6 T(1)*T(4,4)*T(2,1,3)*T(6,3,1) 01(12222)^4(12333)^2(1232323)^6 4,211,113,024 new 74 6 T(5,4)*T(4,1,3)*T(4,3,1) 0(12222)^5(12333)^4(1232323)^4 4,212,19,030 new 78 4 K1,18~K1,25~K1,32 01^{25}12^{18}12^{32} 6,5,4,072 Wang (2005), 3.1.14 80 6 T(1)*T(2,4)*T(1,1,3)*T(9,3,1) 01(12222)^2(12333)(1232323)^9 4,211,119,018 new 80 6 T(3,4)*T(3,1,3)*T(7,3,1) 0(12222)^3(12333)^3(1232323)^7 4,212,115,024 new 81 6 T(6,9)*T(2,1,8) 0(123^8)^2(12^9)^6 4,37,1,063 Wang (2005), p. 68 82 2 K1,81 01^{81} 9,080 WS 86 3 K1,42~K1,42 012^{42}1^{42} 7,6,082 WS 86 6 T(1,12,3,1) 01(2343434)^{12} 4,211,125,012 Yao (2001)? 86 6 T(1,4)*T(2,1,3)*T(10,3,1) 012222(12333)^2(1232323)^{10} 4,212,121,018 new 87 6 T(5)*T(1,9)*T(3,8,1)*T(2,1,8) 01^512^9(1(23)^8)^3(123^8)^2 4,35,2,121,031 new 89 4 K1,31~SK1,28 012^{31}(12)^{28} 6,5,127,031 WS 89 5 T(6)*T(15,4)*T(1,3,1) 01^6(12222)^{15}1232323 5,215,13,051 Wang (2005), Cor 4.1.4(3) 91 4 K1,29~SK1,30 012^{29}(12)^{30} 6,5,129,029 WS 91 6 T(4)*T(3,9)*T(1,5,4)*T(3,1,8) 01^4(12^9)^31(23^4)^5(123^8)^3 4,36,25,067 new 94 4 T(14,4)*T(1,22) 012^{22}(12222)^{14} 5,4,213,064 Yuan (1998)? 95 6 T(5)*T(14,4)*T(1,1,3)*T(2,3,1) 01^5(12222)^{14}(12333)(1232323)^2 5,216,15,051 new 95 6 T(4)*T(15,4)*T(3,1,3) 01^4(12222)^{15}(12333)^3 5,217,1,057 Wang (2005), p. 77 97 4 SK1,48 0(12)^{48} 7,147,0 WS 98 8 T(3,1)*T(4,9)*C(3C(K1+T(7,1))) 0(12)^3(12^9)^4(122(34)^7)^3 4,36,2,123,036 new

The above example on 98 vertices is the smallest integral tree known of diameter 8.

Some version of the above two tables was published in Brouwer (2008).

Let K1,m1~K1,m2~ ... ~K1,mt denote a path with t vertices and mi pending edges at the i-th vertex. Aart Blokhuis remarks that for t=3 and eigenvalues a–1,a,a+1 one gets Pell's equation u2–2v2 = –1 with smallest solutions (m1,m2,m3) = (18,25,32) and (800,841,882).

## Characteristic polynomials

If T is a tree with edge e = xy that splits T into the components A and B with x in A and y in B, and φT is the characteristic polynomial of T, then φT = φAφB – φA\xφB\y

It follows that if G is a tree with designated vertex x, and H a tree with designated vertex y, and a tree T = G~mH is constructed by taking G and m copies of H, where x is joined to the m copies of y, then φT = φHm–1GφH – mφG\xφH\y). By symmetry, if T is integral, and G is integral, then also the result H~mG of taking H and attaching m copies of G is integral.

For example, suppose that T(p,q,r,...) is integral, where p > 1. Then also H = T(q,r,...) is integral and we can apply this with G = K1 to conclude that C(pK1+qT(r,...)) is integral (and much smaller). (For example, the integrality of T(4,9,4) of diameter 6 on 185 vertices is equivalent to that of Watanabe's graph C(9K1,4+4K1) of diameter 4 on 50 vertices. And the integrality of T(5,4) of diameter 4 on 26 vertices is equivalent to that of K1,9 of diameter 2 on 10 vertices.)

This also immediately gives the integrality of Csikvári's trees (see below).

## Traces

One can compute the trace of A2, A4, A6, and finds (with t running over the positive eigenvalues):
∑ t2 = n–1,
∑ t4 = ∑ dx2 – (n–1),
∑ t6 = ∑ dx3 – 3 ∑ dx2 + 2(n–1) + 3 ∑ dxdy
where dx is the degree of the vertex x (so that ∑ dx = 2(n–1)), and the last sum is over unordered edges e = xy.

## Matchings

Theorem (Watanabe (1979)) An integral tree cannot have a complete matching, that is, must have an eigenvalue 0, unless it is K2.

Proof Suppose T is a tree with a complete matching. Then that matching is unique, since the union of two distinct complete matchings contains a cycle. Now the constant term of the characteristic polynomial is, up to sign, the number of complete matchings. It is also the product of all eigenvalues. If this constant term is 1 or –1 and the tree is integral, then all eigenvalues are 1 or –1 and the path of length 2 is not an induced subgraph, so we have K2. ∎

This argument can be extended a little.

Theorem (AEB) If an integral tree has 0 as eigenvalue with multiplicity 1, then the tree is SK1,m.

Proof Suppose T is a tree on n vertices with eigenvalue 0 of multiplicity 1. Then it has almost matchings: coverings by m pairwise disjoint edges and a single point, where n = 2m+1. The number of such almost matchings is, up to sign, the product of the nonzero eigenvalues. On the other hand, the number of such almost matchings is precisely the number of nonzero entries of the unique eigenvector u for 0. (If we delete a vertex where u is zero, then the resulting graph has eigenvalue 0 and hence no matchings. Suppose that u is nonzero at p, and the graph T\p has no matching. Then it has eigenvalue 0, and since n–1=2m is even, this eigenvalue has multiplicity at least 2, so there is an eigenvector v of T\p for 0 that sums to 0 on the neighbours of p. But then v extended by a 0 on p is another eigenvector for 0 of T, contradiction.) This number of nonzero entries is at most (n+1)/2 = m+1, since the nonzero entries form a coclique, and no vertex with zero entry is adjacent to more than two vertices with nonzero entries.

This shows that ∑ t2 = 2m (from the trace of A2) and ∏ t2 ≤ m+1 (from the above), where sum and product are over the positive eigenvalues t. Since T is integral these t are all at least 1, and the extremal situation is when all except one are 1 and the last one has t2 = m+1. Since equality holds we must be in this extremal situation and know the spectrum, it is that of SK1,m.

Now A2–I has rank 3 (eigenvalue m with multiplicity 2 and –1 with multiplicity 1) and hence has rank 1 on one bipartite half of T. The Perron-Frobenius eigenvector is positive everywhere, so yields an eigenvector of A2–I for both components, with eigenvalue m. The vector u vanishes on one bipartite half. That bipartite half is connected for steps of size 2, and has a rank 1 matrix, so has diameter 2, where each vertex has a unique path of length 2 to every other vertex, and has degree 2 itself. This forces the structure, and T must be SK1,m. ∎

Remark The trees SK1,m are determined by their spectrum as trees, not as graphs. For example, SK1,3 is cospectral with C6+K1.

Remark Ghorbani et al. (2016) show that only finitely many integral trees have a given nullity larger than 1. The unique integral tree with nullity 2 is K1,2~K1,2. The unique integral tree with nullity 3 is K1,4.

It is immediately clear that the only integral trees with spectral radii 0 and 1 are K1 and K2. It is easy to check that the only integral trees with spectral radius 2 are the three trees K1,4 and K1,2~K1,2 and SK1,3 (on 5, 6, 7 vertices, respectively). We can go a step further:

Theorem (AEB & WHH) There are 11 integral trees with largest eigenvalue 3 (on 10, 14, 17, 17, 19, 25, 26, 31, 31, 37, 37 vertices, respectively). These trees have already been described above.

## Graham descent

Consider the equation (A2–1)(B2–1) = C2–d, for integers A,B,C,d, where d is nonnegative. Without loss of generality we may assume that A,B,C are nonnegative, and that A ≤ B. Put C = AB–D and E = AD+B. Then the equation given is equivalent to (A2–1)(D2–1) = E2–d of the same shape as the original equation. If A > 1 and d < 27, then |D| < B, and we have found a strictly smaller solution.

Apply this argument to the characteristic equations of K1,m~K1,r and K1,m~SK1,r The unknown eigenvalues a, –a, b, –b satisfy a2+b2 = m+r+1 and a2b2 = mr in the first case, and a2+b2 = m+r+2 and a2b2 = mr + m + 1 in the second case.

With A = b–a and B = b+a these are equivalent to m,r = (A2+B2–2±2C)/4 where C2 = (A2–1)(B2–1) in the first case, and to m–1,r = (A2+B2–6±2C)/4 where C2–4 = (A2–1)(B2–1) in the second case.

By Graham descent we arrive after a number of steps at a solution with A ≤ 1. Now A = 0 does not give solutions with positive m,r, and A = 1 leads to B = 2a+1 and C = 0 and m,r = a(a+1) in the first case, and to B = 2a+1 and C = 2 and {m–1,r} = {a(a+1),a(a+1)–2}, that is, (m,r) = (a(a+1)+1,a(a+1)–2) or (a(a+1)–1,a(a+1)) in the second case.

For the first case Graham (1980) gives an explicit expression of A,B,C in terms of Chebyshev polynomials.

## Large diameter

Csikvári (2010) constructs trees somewhat similar to the trees T(nk,...,n1), let us call them T'(r1,...,rm). These are defined by induction as follows: T'() is the tree with a single vertex x0. T'(r1,...,rm) is the tree obtained from T'(r1,...,rm–1) by adding rm pending edges to each vertex that has a distance to x0 that has the same parity as m–1. The diameter of this tree is 2m (assuming r1 > 1), and it has 2m+1 distinct eigenvalues. The spectrum of this tree can be given explicitly. In particular, T'(a12–a22, a22–a32, ..., am–12–am2, am2) has integral nonnegative eigenvalues a1, ..., am, 0.

Integrality of these trees can also be seen using the above remark on characteristic polynomials. It implies that if T'(a+b,c,d,...) and T'(b,c,d,...) and T'(c,d,...) are integral, then also T'(a,b,c,d,...) is.

The trees T'(r,m) are the same as T(r,m). The trees T'(5,3,1) and T'(12,3,1) are the same as the trees T(1,5,3,1) and T(1,12,3,1) seen above. More generally, the trees T'(r,m,t) are the same as the trees T(t)*T(r,m,t) studied by Wang & Li (2000).

This yields integral trees of arbitrary even diameter. For example, an integral tree of diameter 12 on 27007 vertices. The case of large odd diameter was settled by Ghorbani et al. (2010) using a variation on this construction.

## References

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Ligong Wang, Xueliang Li & Shenggui Zhang, Families of integral trees with diameters 4, 6, and 8, Discrete Applied Mathematics 136 (2004) 349–362. pdf

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