General degree-eccentricity index of unicyclic graphs of given order, girth and maximum degree

For a connected graph G and a, b is an element of R, the general degree-eccentricity index of G is defined as DEIa,b(G) = & sum;(v is an element of V(G) )d(G)(a)(v)ecc(G)(b)(v), where V(G) is the vertex set of G, d(G)(v) is the degree of a vertex v and ecc(G)(v) is the eccentricity of v in G, i.e. the maximum distance from v to another vertex of the graph. This index generalizes several well-known 'topological indices' of graphs such as the eccentric connectivity index. We characterize the unique unicyclic graphs with the maximum and the minimum general degree-eccentricity index among all n-vertex unicyclic graphs with fixed order, girth, and maximum degree for the cases a >= 1, b <= 0 and 0 <= a <= 1, b >= 0. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).