A NOTE ON FAMILIES OF INTEGRAL TREES

OF DIAMETER 4, 6, 8, AND 10Pavel Híc, Milan Pokorný

Katedra matematiky a informatiky

Pedagogická fakulta

Trnavská univerzita v Trnave

A graph G is called integral, if it has an integral spectrum, i.e. if all zeros of the characteristic polynomial P(G;x) are integers.

Note.G - graph.A(G) - adjacency matrix of G.P(G; x)=x.In – A(G).

Definition. (Harary and Schwenk -1974).

The general question: For what classes of graphs is it possible to characterize all the graphs that are integral?

1. It is known (Cvetkovič and Bussemaker 1976) that there are exactly 13 integral cubic graphs.

2. Bipartite, nonregular integral graphs of maximum degree 4 have been studied by Baliňska and Simič 2001. For example, it is shown that no graph in this class has more than 78 vertices.

3. Trees present another important family of graphs for which the problem has been considered. Specially, balanced trees have been studied by Nedela, Híc, Schwenk, Watanabe, and others .

Results:

Balanced Integral Trees of Even Diameter.

Every balanced tree T of diameter 2k is defined by a sequence (nk, nk-1,..., n1) and denoted by T(nk, nk-1,..., n1).

T(4): T(45,4):

4. Definition. A tree T is called a balanced tree, if the vertices at the same distance from the centre of T have the same degree.

}45

T(16,45,4):

Balanced Integral Trees of Even Diameter.

45

}16

45

45

Theorem 1.(Schwenk, Watanabe, 1979)T(n1) is integral if and only if n1=t2.Theorem 2.(Schwenk, Watanabe, 1979)T(n2 , n1) is integral if and only if n1=t2, n2=n2+2nt.Theorem 3. (Nedela, Híc, 1998)T(n3, n2, n1) is integral if and only if

n1=t2, n2= n2+2nt, n3 =

where a, b, n, t are positive integers satisfying (t2-b2)(a2-t2) = t2(n2+2nt), b t a.

Results:

2

22

t

ba

Theorem 4. (Nedela, Híc, 1998)T(n4, n3, n2, n1) ) is integral if and only if n1=t2, n2= n2+2nt,

n3= n4= where a, b, c, d, t, n are positive integers satisfying (t2-b2)(a2-t2) = t2(n2+2nt), b t a.and(c2+d2)(n+t)2t2=(n+t)4t2 + a2b2(n2+2tn) + c2d2t2, a2b2 c2d2.

Theorem 5.(Nedela, Híc, 1998)Let T(nr, nr-1, ..., n2, n1) ) be integral. Then T(nr-1, nr-2,..., n2, n1), ..., T(n4, n3, n2, n1) , T(n3, n2, n1), T(n2, n1), T(n1) are integral.

Results:

2

22

t

ba2

2222

)tn(

badc

A number of integral balanced trees T(n2, n1) for n1 = t2, n2 = n2+2nt, n,t 1...1000, is 1 000 000.

A number of integral balanced trees T(n3, n2, n1) for n1 = t2, n2 = n2+2nt, n,t 1...1000, is 35 061.

A number of integral balanced trees T(n4, n3, n2, n1) for n1 = t2, n2 = n2+2nt, n,t 1...1000, is 5 062.

A number of integral balanced trees T(n5, n4, n3, n2, n1) for n1 = t2, n2 = n2+2nt, n,t 1...1000, is 2.

Results: Integral Balanced Trees of Diameter 10 (Híc, Pokorný 2002)

Theorem 6. (Híc, Pokorný 2002):T(3 006 756, 1 051 960, 751 689, 283 360, 133 956) is integral and its spectrum Sp(T) = {0, 289, 306, 366, 527, 646, 918, 1 037, 1 394, 2 074}.

Corollary:T(3 006 756 q2, 1 051 960 q2, 751 689 q2, 283 360 q2, 133 956 q2) is integral for every q N and its spectrum Sp(T) = {0, 289 q, 306 q, 366 q, 527 q, 646 q, 918 q, 1 037 q, 1 394 q, 2 074 q}.

Results: Integral Balanced Trees of Diameter 10

These authors investigate trees T(n1)*T(k3,k2,k1) and T(n1)*T(k4,k3,k2,k1), created by identifying the center w of T(n1) with the center u of either T(k3,k2,k1), or T(k4,k3,k2,k1).

Note:Ligong Wang, Xueliang Li, Shenggui Zhang (DISCRETE APLIED MATHEMATICS 2004)

1. If G=T(k2,k1) of diameter 4 and T(n1)*T(k2,k1) of diameter 4 are integral, then T(n1,k2,k1) of diameter 6 is integral.

2. If T(k3,k2,k1) of diameter 6 is integral and T(n1)*T(k3,k2,k1) of diameter 6 is integral, then T(n1,k3,k2,k1) of diameter 8 is integral.

Results: Ligong Wang, Xueliang Li, Shenggui Zhang (DISCRETE APLIED MATHEMATICS 2004)

Are there any integral trees

T(n1)*T(k4,k3,k2,k1),

T(n2,n1)*T(k3,k2,k1),

T(n2,n1)*T(k4,k3,k2,k1) and so on?

Questions: Ligong Wang, Xueliang Li, Shenggui Zhang (DISCRETE APLIED MATHEMATICS 2004)

1.If T(nk,nk-1,...,n2,n1) is integral then T(nk)*T(nk-1,nk-2,...,n2,n1)is integral.

2. If T(nk-1,nk-2,...,n2,n1) and T(nk)*T(nk-

1,nk-2,...,n2,n1) are integral then T(nk,nk-1,...,n2,n1) is integral too.

Theorems (Híc, Pokorný).

3.If T(nk,nk-1,...,n2,n1) is integral and nk is a perfect square then T(nk-1,nk)*T(nk-2,nk-3,...,n2,n1)is integral.

Theorems (Híc, Pokorný).

Corollaries (Híc, Pokorný).

1. For every integral tree T(n5,n4,n3,n2,n1) the tree T(n5)*T(n4,n3,n2,n1) is integral too.

2. For every integral tree T(n5,n4,n3,n2,n1) where n5 is a perfect square the tree T(n4,n5)*T(n3,n2,n1) is integral too.

Corollaries (Híc, Pokorný).

Using Theorem 6. (Híc Pokorný 2002):

3. The tree T(3 006 756)*T(1 051 960, 751 689, 283 360, 133 956) is integral.

4. The tree T(1 051 960, 3 006 756)*T(751 689, 283 360, 133 956) is integral.

Notes (Híc, Pokorný).

There are many integral trees which belong either to the class T(n5)*T(n4,n3,n2,n1), or to the class T(n4,n5)*T(n3,n2,n1), but the corresponding tree T(n5,n4,n3,n2,n1) is not integral.

There are 10560 trees which belong to the class T(n4,n5)*T(n3,n2,n1) for n1,n3,n4 ,n5,n1+n2 < 1001. 9480 of them are primitive.

Thank you for your attention.