General Multiplicative Zagreb Indices of Graphs with a Small Number of Cycles

Abstract

We present lower and upper bounds on the general multiplicative Zagreb indices for bicyclic graphs of a given order and number of pendant vertices. Then, we generalize our methods and obtain bounds for the general multiplicative Zagreb indices of tricyclic graphs, tetracyclic graphs and graphs of given order, size and number of pendant vertices. We show that all our bounds are sharp by presenting extremal graphs including graphs with symmetries. Bounds for the classical multiplicative Zagreb indices are special cases of our results.

Quasi-Tree Graphs With Extremal General Multiplicative Zagreb Indices

“…Let Q Q Q 2 n,k be the n-vertex quasi-tree graphs having k pendant vertices and degree sequence (n − k − Figure 3, we have drawn Q Q Q 2 7,2 and Q Q Q 2 10,5 . Obviously, sometimes the graph Q Q Q 2 n,p is not unique.…”

“…Vetrík and Balachandran [18] determined the minimal and maximal general multiplicative Zagreb indices for trees with fixed order or segments or branching vertices or number of pendant vertices, and they also identified the extremal trees. Other relevant conclusions on general multiplicative Zagreb indices can be found in [1][2][3]17]. We only deal with the simple connected graphs in this work.…”

General multiplicative Zagreb indices of unicyclic graphs

“…Lower and upper bounds on the multiplicative Zagreb indices for unicyclic graphs of given order were obtained by Xu and Hua [15], sharp upper bounds for graphs of prescribed order and size were obtained in [7], sharp upper bounds for bipartite graphs of given diameter were given in [11], trees were investigated in [5] and [12], k-trees in [14], graphs with a small number of cycles in [1], graph operations in [4] and [8], graphs with cut edges in [13], graphs with given chromatic number in [3], graphs with prescribed clique number in [9], some derived graphs in [2] and molecular graphs in [6].…”

General Multiplicative Zagreb Indices of Graphs With Bridges

“…Tight lower and upper bounds on the multiplicative Zagreb indices for graphs with given number of vertices and bridges were given in [14], sharp upper bounds for graphs with given order and size were obtained in [9], bounds for graphs with respect to order and clique number wer given in [12], tight lower and upper bounds for trees, unicyclic graphs and bicyclic graphs of given order were presented in [15], upper bounds for graph products were obtained in [4], lower bounds for graph operations were investigated in [11], graphs of given order and chromatic number in [16], graphs with a small number of cycles in [1], derived graphs in [2], molecular graphs in [6] and upper bounds for bipartite graphs were studied in [13]. Classical Zagreb indices were investigated in [8] and [10], the augmented Zagreb index in [3] and [7], and weighted Harary indices for graphs with bridges in [5].…”