A new expression for the adjoint polynomial of a path

For k ≥ 3, let G = T (l1, l2, . . . , lk) be the tree with exactly one vertex v of degree k, and G−v = Pl1∪Pl2∪· · ·∪Plk , where Pn is the path of order n. ∗ Supported by the National Science Foundation of China (No. 10761008). † Corresponding author. Second address: Department of Mathematics and Information Science, Qinghai Normal University, Xining, Qinghai 810008, P. R. China. 104 J. WANG, K.L. TEO, Q.X. HUANG, C. YE AND R. LIU A graph is adjointly integral if all the roots of its adjoint polynomial are integers. In this paper, we derive a new expression (different from the one given by Dong, Teo, Little and Hendy in Australas. J. Combin. 25 (2002), 167–174) for the adjoint polynomial of a path to provide a new method for finding the roots of the adjoint polynomials of paths and cycles. Moreover, we establish the following results: 1. k 2 1−k < β(G) ≤ −k, where β(G) is the minimum real root of the adjoint polynomial of G. 2. The tree T (l, l, . . . , l) is adjointly integral if and only if l ∈ {1, 2}. 3. If l1 ≥ 12 or 1 ≤ li ≤ 2 and (l1, l2, . . . , lk) ∈ {(1, 1, . . . , 1), (2, 2, . . . , 2)}, then T (l1, l2, . . . , lk) is not adjointly integral for l1 ≥ l2 ≥ · · · ≥ lk ≥ 1.