Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

Abstract

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

Symmetric Division Deg Index: Extremal Results and Bounds

“…Also, in [75], the graphs having the maximum SDD index were characterized over the classes of all (i) connected unicyclic graphs possessing perfect matching (ii) connected bicyclic graphs possessing perfect matching (iii) n-order connected unicyclic graphs with a fixed matching number (iv) n-order connected bicyclic graphs with a fixed matching number. Recently, the problem of finding the graphs attaining the first five minimum values of the SDD index among all connected bicyclic graphs possessing perfect matching was attacked in [71]. The problem of characterizing the graphs having the maximum SDD index in the class of all connected molecular bicyclic graphs possessing perfect matching was also addressed in [71].…”

“…Recently, the problem of finding the graphs attaining the first five minimum values of the SDD index among all connected bicyclic graphs possessing perfect matching was attacked in [71]. The problem of characterizing the graphs having the maximum SDD index in the class of all connected molecular bicyclic graphs possessing perfect matching was also addressed in [71].…”