| Molecular graph invariants are often applied to study molecular,chemistry in the quantitative structure-property relationship(QSPR)and the quantitative structure-activity relationship(QSAR)analysis.They are also often used to describe the physical and chem-ical properties of compounds,such as the boiling point,melting point,heat capacity,tox-icity,and enthalpy of evaporation,etc.Topological index is a typical molecular graph invariant,which is a molecular descriptor calculated based on the molecular structure graph.Different graph invariants(topological indices)can be constructed based on dif-ferent types of graph parameters.In this paper,we mainly study the extremal problems of several spectrum-based and degree-based graph invariants(topological indices).The specific content is as follows:The extremal problems of the extended energy and the geometric-arithmetic Estrada index.First,we present some inequalities about the extended spectral radius in terms of some invariant graph parameters and some other topological indices.Some spectrum properties of the extended adjacency matrix are characterized,and the Koolen-Moulton bounds of the extended energy are presented.In addition,we obtain the explicit for-mula of the geometric-arithmetic spectral moment,and propose some sharp bounds of the geometric-arithmetic Estrada index in(general)graphs,bipartite graphs and regular graphs.Finally,the connections between the geometric-arithmetic Estrada index and the geometric-arithmetic energy are given.The extremal Randi(?)-type indices of trees with a given domination number.The upper bound of the general Randi(?)index R_αof trees forα∈[-0.5287,0)with a given domination number,and the lower bound of the general Randi(?)index R_αof trees forα∈[-0.5696,0)with a given domination number are obtained.The bounds of the zeroth-order general Randi(?)index~0R_αof trees forα∈(0,1)andα∈(-∞,0)∪(1,∞)with a given domination number are also proposed.Meanwhile,the corresponding extremal trees are characterized. |
