On the Minimum Real Roots of the Adjoint Polynomials of Graphs

Abstract:

      In this paper, we are concerned with the minimum real root of the adjoint polynomial of the connected graph $G$ with cut-vertex $u$, in which $G-u$ contains paths, circles or $D_{n}$ components. Here $D_{n}$ is the graph obtained from $K_{3}$ and path $P_{n-2}$ by identifying a vertex of $K_{3}$ with an end-vertex of $P_{n-2}$. Some relevant ordering relations are obtained. This extends several previous results on the minimum roots of the adjoint polynomials of graphs.