| Abstrakt: |
Let T be a tree with vertex set V (T) = { v 1 , v 2 , ... , v n }. The adjacency matrix A (T) of T is an n × n matrix (a i j) , where a i j = a j i = 1 if v i is adjacent to v j and a i j = 0 if otherwise. In this paper, we consider the multiplicity of − 1 as an eigenvalue of A (T) , which is written as m (T , − 1). It is proved that among all trees T with p ≥ 2 pendant vertices, the maximum value of m (T , − 1) is p−1, and for a tree T with p ≥ 2 pendant vertices, m (T , − 1) = p − 1 if and only if T = P n with n ≡ 2 (m o d 3) , or T is a tree in which d (v , u) ≡ 2 (m o d 3) for any pendant vertex v and any major vertex u of T, where a major vertex is a vertex of degree at least 3 and d (v , u) is the distance between v and u. [ABSTRACT FROM AUTHOR] |
