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

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.I n – 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 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 (n k, n k-1,..., n 1 ) and denoted by T(n k, n k-1,..., n 1 ). 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

Theorem 1.(Schwenk, Watanabe, 1979) T(n 1 ) is integral if and only if n 1 =t 2. Theorem 2.(Schwenk, Watanabe, 1979) T(n 2, n 1 ) is integral if and only if n 1 =t 2, n 2 =n 2 +2nt. Theorem 3. (Nedela, Híc, 1998) T(n 3, n 2, n 1 ) is integral if and only if n 1 =t 2, n 2 = n 2 +2nt, n 3 = where a, b, n, t are positive integers satisfying (t 2 -b 2 )(a 2 -t 2 ) = t 2 (n 2 +2nt), b  t  a. Results:

Theorem 4. (Nedela, Híc, 1998) T(n 4, n 3, n 2, n 1 ) ) is integral if and only if n 1 =t 2, n 2 = n 2 +2nt, n 3 = n 4 = where a, b, c, d, t, n are positive integers satisfying (t 2 -b 2 )(a 2 -t 2 ) = t 2 (n 2 +2nt), b  t  a. and (c 2 +d 2 )(n+t) 2 t 2 =(n+t) 4 t 2 + a 2 b 2 (n 2 +2tn) + c 2 d 2 t 2, a 2 b 2  c 2 d 2. Theorem 5.(Nedela, Híc, 1998) Let T(n r, n r-1,..., n 2, n 1 ) ) be integral. Then T(n r-1, n r-2,..., n 2, n 1 ),..., T(n 4, n 3, n 2, n 1 ), T(n 3, n 2, n 1 ), T(n 2, n 1 ), T(n 1 ) are integral. Results:

A number of integral balanced trees T(n 2, n 1 ) for n 1 = t 2, n 2 = n 2 +2nt, n,t  , is A number of integral balanced trees T(n 3, n 2, n 1 ) for n 1 = t 2, n 2 = n 2 +2nt, n,t  , is A number of integral balanced trees T(n 4, n 3, n 2, n 1 ) for n 1 = t 2, n 2 = n 2 +2nt, n,t  , is A number of integral balanced trees T(n 5, n 4, n 3, n 2, n 1 ) for n 1 = t 2, n 2 = n 2 +2nt, n,t  , is 2. Results: Integral Balanced Trees of Diameter 10 (Híc, Pokorný 2002)

Theorem 6. (Híc, Pokorný 2002): T( , , , , ) is integral and its spectrum Sp(T) = {0,  289,  306,  366,  527,  646,  918,  1 037,  1 394,  2 074}. Corollary: T( q 2, q 2, q 2, q 2, q 2 ) is integral for every q  N and its spectrum Sp(T) = {0,  289 q,  306 q,  366 q,  527 q,  646 q,  918 q,  q,  q,  q}. Results: Integral Balanced Trees of Diameter 10

These authors investigate trees T(n 1 )*T(k 3,k 2,k 1 ) and T(n 1 )*T(k 4,k 3,k 2,k 1 ), created by identifying the center w of T(n 1 ) with the center u of either T(k 3,k 2,k 1 ), or T(k 4,k 3,k 2,k 1 ). Note:Ligong Wang, Xueliang Li, Shenggui Zhang (DISCRETE APLIED MATHEMATICS 2004)

1. If G=T(k 2,k 1 ) of diameter 4 and T(n 1 )*T(k 2,k 1 ) of diameter 4 are integral, then T(n 1,k 2,k 1 ) of diameter 6 is integral. 2. If T(k 3,k 2,k 1 ) of diameter 6 is integral and T(n 1 )*T(k 3,k 2,k 1 ) of diameter 6 is integral, then T(n 1,k 3,k 2,k 1 ) of diameter 8 is integral. Results: Ligong Wang, Xueliang Li, Shenggui Zhang (DISCRETE APLIED MATHEMATICS 2004)

Are there any integral trees T(n 1 )*T(k 4,k 3,k 2,k 1 ), T(n 2,n 1 )*T(k 3,k 2,k 1 ), T(n 2,n 1 )*T(k 4,k 3,k 2,k 1 ) and so on? Questions : Ligong Wang, Xueliang Li, Shenggui Zhang (DISCRETE APLIED MATHEMATICS 2004)

1.If T(n k,n k-1,...,n 2,n 1 ) is integral then T(n k )*T(n k-1,n k-2,...,n 2,n 1 )is integral. 2. If T(n k-1,n k-2,...,n 2,n 1 ) and T(n k )*T(n k-1,n k-2,...,n 2,n 1 ) are integral then T(n k,n k-1,...,n 2,n 1 ) is integral too. Theorems (Híc, Pokorný).

3.If T(n k,n k-1,...,n 2,n 1 ) is integral and n k is a perfect square then T(n k-1,n k )*T(n k-2,n k-3,...,n 2,n 1 )is integral. Theorems (Híc, Pokorný).

Corollaries (Híc, Pokorný). 1.For every integral tree T(n 5,n 4,n 3,n 2,n 1 ) the tree T(n 5 )*T(n 4,n 3,n 2,n 1 ) is integral too. 2.For every integral tree T(n 5,n 4,n 3,n 2,n 1 ) where n 5 is a perfect square the tree T(n 4,n 5 )*T(n 3,n 2,n 1 ) is integral too.

Corollaries (Híc, Pokorný). Using Theorem 6. (Híc Pokorný 2002): 3. The tree T( )*T( , , , ) is integral. 4. The tree T( , )* T( , , ) is integral.

Notes (Híc, Pokorný). There are many integral trees which belong either to the class T(n 5 )*T(n 4,n 3,n 2,n 1 ), or to the class T(n 4,n 5 )*T(n 3,n 2,n 1 ), but the corresponding tree T(n 5,n 4,n 3,n 2,n 1 ) is not integral. There are trees which belong to the class T(n 4,n 5 )*T(n 3,n 2,n 1 ) for n 1,n 3,n 4,n 5,n 1 +n 2 < of them are primitive.

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