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A NOTE O N FAMILIES OF INTEGRAL TREES OF DIAMETER 4, 6, 8, AND 10

A NOTE O N FAMILIES OF INTEGRAL TREES OF DIAMETER 4, 6, 8, AND 10. Pav e l Híc, Milan Pokorný Katedra matematiky a  informatiky Pedagogická fakulta Trnavská univerzita v Trnav e. Definition. ( Harary and Schwenk - 1974).

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A NOTE O N FAMILIES OF INTEGRAL TREES OF DIAMETER 4, 6, 8, AND 10

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  1. A NOTE ON FAMILIES OF INTEGRAL TREES OF DIAMETER 4, 6, 8, AND 10 Pavel Híc, Milan Pokorný Katedra matematiky a informatiky Pedagogická fakulta Trnavská univerzita v Trnave

  2. Definition.(Harary and Schwenk-1974). 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).

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

  4. }45 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. 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):

  5. 45 45 45 }16 Balanced Integral Trees of Even Diameter. T(16,45,4):

  6. Results: 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.

  7. Results: 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.

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

  9. Results:Integral Balanced Trees ofDiameter 10 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,  306q,  366q,  527q,  646q,  918q,  1 037q,  1 394q,  2 074q}.

  10. Note:Ligong Wang, Xueliang Li, Shenggui Zhang (DISCRETE APLIED MATHEMATICS 2004) These authors investigate treesT(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), orT(k4,k3,k2,k1).

  11. Results: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.

  12. Questions: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?

  13. Theorems (Híc, Pokorný). 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.

  14. 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.

  15. Corollaries (Híc, Pokorný). • For every integral tree T(n5,n4,n3,n2,n1) the tree T(n5)*T(n4,n3,n2,n1) is integral too. • 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.

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

  17. 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.

  18. Thank you for your attention.

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