Complete solution to a conjecture of Zhang-Liu-Zhou

‎‎Let d_{n,m}=\big[\frac{۲n+۱-\sqrt{۱۷+۸(m-n)}}{۲}\big] and‎ ‎E_{n,m} be the graph obtained from a path‎ ‎P_{d_{n,m}+۱}=v_۰v_۱ \cdots v_{d_{n,m}} by joining each vertex of‎ ‎K_{n-d_{n,m}-۱} to v_{d_{n,m}} and v_{d_{n,m}-۱}‎, ‎and by‎ ‎joining m-n+۱-{n-d_{n,m}\choose ۲} vertices of K_{n-d_{n,m}-۱}‎ ‎to v_{d_{n,m}-۲}‎. ‎Zhang‎, ‎Liu and Zhou [On the maximal eccentric‎ ‎connectivity indices of graphs‎, ‎Appl‎. ‎Math‎. ‎J‎. ‎Chinese Univ.‎, ‎in‎ ‎press] conjectured that if d_{n,m}\geqslant ۳‎, ‎then E_{n,m}‎ ‎is the graph with maximal eccentric connectivity index among all‎ ‎connected graph with n vertices and m edges‎. ‎In this note‎, ‎we‎ ‎prove this conjecture‎. ‎Moreover‎, ‎we present the graph with‎ ‎maximal eccentric connectivity index among the connected graphs‎ ‎with n vertices‎. ‎Finally‎, ‎the minimum of this graph invariant‎ ‎in the classes of tricyclic and tetracyclic graphs are computed‎.

Keywords:

Eccentric connectivity index , tricyclic graph , tetracyclic graph , graph operation