The Difference of Zagreb Indices of Halin Graphs

The difference of Zagreb indices of a graph G is defined as ΔM(G)=∑u∈V(G)(d(u))2−∑uv∈E(G)d(u)d(v), where d(x) denotes the degree of a vertex x in G. A Halin graph G is a graph that results from a plane tree T without vertices of degree two and with at least one vertex of degree at least three such that all leaves are joined through a cycle C in the embedded order. In this paper, we establish both lower and upper bounds on the difference of Zagreb indices for general Halin graphs and some special Halin graphs with fewer inner vertices. Furthermore, extremal graphs attaining related bounds are found.