Bounds For The Energy Of Graphs And Energy Ordering Of Unbalanced Bicyclic Signed Graphs With Two Cycles Of Length 4

In 1736,graph theory originated in the 18th century,marked by the event "Konigsberg seven Bridges Problem".Euler,a mathematician,is known as the father of graph theory.At present,graph theory has been widely used to solve many practical problems in nature.The study of graph energy is also extended from undirected graph to directed graph,mixed graph,symbolic graph and so on.The paper,firstly some new upper and lower bounds of graph energy are obtained by using the number of vertices n,edges m,spectral radius p and minimum eigenvalue r of graph.Next we give that energy ordering of unbalanced bicyclic signed graphs with two cycles of length 4.The energy of a particular signed graph is discussed,such as Bn-(i,i)?Ln-(4,4)?Mn-(4,4).Among them,the shape of double circle symbol graph is also discussed.The main methods used are quasi-order method,new method of directly comparing the energies of two subdivision bipartite signed graphs and so on.We have if n?23,there is the unblanced bicyclic signedgraphs of order n with two cycles of length 4 from the first largest to the(?)th largest energies.U4,4-(0,n-9)(?)U4,4-(2,n-11)(?)…U4,4-(2t,n-9-2t)(?)U4,4-(2l+1,n-9-2l-1)(?)…(?)U4,4-(5,n-14)(?)U4,4-(3,n-12)(?)Ln-(4,4)(?)Mn-(4,4).where,(?)...