Some Results On Extremal Matching Energy Of Graphs

Matching energy of a graph was introduced by Gutman and Wagner in 2012, it can be defined as the sum of absolute values of zeros of its matching polynomial. It coincides with the Coulson-type integral formula for the energy when the graph under consideration is a tree. Based on Parameter Control, this paper apply some properties of the matching energy to study connected graphs, uncyclic graphs, tricyclic graphs.Firstly, in the second chapter of this paper, We characterize the unicyclic graphs of order n with fixed girth and extremal(i.e., maximum and minimum) matching energy. And we prove its uniqueness. Let n, g be positive integers, n> g≥3. For any connected graph G ∈ug,n, ME(Cg(Sn-g+1))< ME(G)≤ME(Eg,n), we have with equality if and only if G≌Cg(Sn-g+1) and G≌Eg,n, respectively.Secondly, in the third chapter of this paper,the extremal graphs among all connected graphs and graphs with given order and clique number are characterized. And we prove its uniqueness. using the computer simulation method, the extremal graphs Tχ,n among all connected graphs with given order and clique number are characterized. Among all graphs with clique number l and order n, it is easily to verify that the graph Kl U En-l, where En-l is the empty graph on n-l vertices, uniquely attains minimum matching energy. For any graph G∈ωn,l, we have ME(Ki(Sn-l+1))≤ ME(G)< ME(Ti,n) with equality if and only if G≌Ki(Sn-i+1) and G≌Tl,n, respectively.Finally, in the fourth chapter of this paper,by some properties of the matching energy, we characterize graphs that attain the maximum matching energy among all connected tricyclic graphs of order n with three vertex-disjoint cycles of length 6. For any graph G∈G6,n\{Φ6Ⅱ(n-17,2,2)} with n≥19, we have ME(G)< ME(Φ6Ⅱ(n-17,2,2)) and, moreover, G(?)Φ6Ⅱ(n-17,2,2) except when n=20,22. We would like to add that when n= 18, G6,nΠ is empty, G6,nⅠ=Φ6,nⅠ and Φ6,nⅠ(2,2;2)(?)G for every G∈G6,n\Φ6,n≈(2,2;2)-When n<18,G6,n is empty.