Graphs With Extremal Values Of The Multiplicative Atom-bond Connectivity Index

The topological indices of molecules graphs is an important field in chemistry graph theory.The topological indices is an invariant of a graph which is directly generated in the molecular structure and reflects the structural characteristics of the compound.In 1947,H.Wiener,a physical chemist in the United States,introduced the Wiener index which the first recognized molecular topological index in the chemical industry,and led to the rapid development of molecular topology.For a new topological indices,people main studies that calculate the topological indices of a graph,the graphs which maximum and minimum value of topological index in some special graphs and some special graph classes according to the topological index of sorting,etc.In 2016,Kulli defined a new topological index called the multiplicative atom-bond connectivity index?in short,the M-ABC index?.The multiplicative atom-bond connectivity index of a graph G can be formulated by M-ABC?G?=???where E?G?denotes the edge set of graph G,du denotes the degree of vertex u.At present,the values of M-ABC index which only several nanotube structures and common drugs were calculated in all the literature about M-ABC index.However,structure graph which correspond to the extreme value of the M-ABC index has not been determined.The maximum?minimum?values of the M-ABC index in all connected simple graphs?Fixed vertex number?and the structural features of the tree with the minimum value of the M-ABC index are studied in this article.Following conclusions are obtained and proved:?1?Let G be a connected simple graph with n vertices different than a star graph.Let x1x2 be an edge of G,where G-x1x2 has no isolate edge.Then M-ABC?G?<M-ABC?G-x1x2?.?2?For the simple connected graph of n vertices,a graph of the minimum M-ABC index is complete graph Kn.?3?Let T be a tree with n?n>3?vertices,where T is not the star graph S_n-1.ThenM-ABC?T?<M-ABC(S_n-1).?4?For the simple connected graph of n vertices,a graph of the maximum M-ABC index is star graph S_n-1.?5?The n-vertex tree with minimal M-ABC index does not contain internal paths of length k?k ? 2?if n?10;The n-vertex tree with minimal M-ABC index does not contain pendent paths of length k?k ? 4?if n ? 10.The n-vertex tree with minimal M-ABC index contains at most one pendent path of length 3 if n ? 10;...