Integral starlike trees

Abstract

In this note we determine which of the trees homeomorphic to a star have a spectrum consisting entirely of integers. We also specify the integral double stars, and we consider the problem of trees with more complicated structure.Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 05.

On Some Properties of Pseudo-Semiregular Graphs

“…During the past few decades, for certain families of integral graphs, many novel results have been obtained [4,6,[21][22][23][24][25][26][27][28][29]. For the construction of integral trees, some efficient methods have been developed [6,22,[25][26][27][28][29].…”

“…The completely full-degree tree T 5,3 depicted in Fig.11. is isomorphic to the strongly balanced tree T(5,5) which is an integral graph [22] with the spectrum Spec(T(5,5))= {3, 2 4 , 0 16 , -2 4 , -3}. It is worth noting that among strongly balanced trees there are several integral graphs.…”

Integer index in trees of diameter 4

“…Watanabe and Schwenk, in [22], have determined the eigenvalues of a tree R(0, r, a), with r ≥ 2 and a ≥ 1, proving that it is integral if and only if a and a + r are perfect squares. In fact, from [1], we have that the positive eigenvalues of R(b, r, a) are √ a and the positive roots of the equation…”

Some constructions of integral graphs

“…Notice that in all theorems in this section, if we put n = 1, then we find some integral graphs if and only if the other parameters satisfy only the second numerical condition. For example, the stars K 1, t 2 for any positive integer t and the graphs obtained in Theorem 8 whenever n = r = 1 consist the only integral trees with exactly one vertex of degree more than two [18].…”