Balanced integral trees (Q2702772)

A graph \(G\) is called integral if all the zeros of its characteristic polynomial \(P(G,\lambda)\) are integers. A tree \(T\) is said to be balanced if all vertices of the same distance from the centre \(Z(T)\) of \(T\) are of the same degree. The authors give a survey of known results and present some new results on balanced integral trees. They prove that there are infinitely many balanced integral trees of diameter \(8\), however there is no balanced integral tree of diameter \(7\) and \(4k+1\) for \(k\geq 1\).