Lower Bounds Of Graph Energy In Terms Of Matching Number

The energye(G) of a graph G is the sum of the absolute values of all eigenvalues of G,denote by?(G)=(?).Let V(G)and E(G)denote the vertex set and edge set of G.A matching M in G is a set of pairwise non-adjacent edges,that is,no two edges in M share a common vertex.A maximum matching is a matching that contain the largest possible number of edges.The matching number of G,denoted by ?(G).We are interested in the relation between the energy of a graph G and the matching number ?(G) of G.For every graph G,it is proved that ?(G)?2?(G),and ?(G)?2?(G)+(?)c1(G)if the cycles of(if any)are pairwise vertex-disjoint,where)(1Gc denotes the number of odd cycles in G.Besides,we prove that ?(G)?r(G)+1/2if G has at least one odd cycle and it is not of full rank.The aim of this paper is study lower bounds of graph energy in terms of matching number.The paper is divided into five chapters.In Chapter 1,the author introduces the background and the development of the subject chosen.Besides these,the main symbolic knowledge of this thesis are explained.In Chapter 2,we introduce some definitions and known results related to our topic.In Chapter 3,the author studies lower bounds of graph energy in terms of rank.In Chapter 4,the author proves lower bounds of graph energy in terms of matching number.In Chapter 5,core conclusions are summarized,accompanied with the direction of the future research.