Research On The Energy Of Graphs With Few Distinct Eigenvalues

Graphs with few distinct eigenvalues s（3≤s≤5）,such as strongly regular graphs and distance regular graphs,typically have special combinatorial properties and symmetry,which have important theoretical significance and research value in graph theory and coding theory.The energy of a graph is defined as the sum of the absolute values of the eigenvalues of the adjacency matrix of it.If the energy of two graphs is equal,they are called equienergetic.If the energy of a graph and its complement is equal,the graph is called complementary equienergetic.Graph energy is an important research direction in chemical graph theory,This article mainly studies the analytical expression of graph energy with 3 and 4 different eigenvalues and the construction of equienergetic graphs.The specific content is as follows:Firstly,we provide the energy analytical expression of a strongly regular graph through its parameters v,k,λ,μ,and then provide a necessary and sufficient condition for complementary equienergetic strongly regular graph.Secondly,regarding to strongly regular graphs SRG（v,k,λ,λ）,we prove that the necessary and sufficient condition for they are complementary equienergetic is that their parameters are（4n²,2n²-n,n²-n,n²-n）.The above conclusion covers a result of Remane et al.For 2≤n≤17,we have determined the parameters of all complementary equienergetic strongly regular graphs that satisfy the necessary and sufficient conditions mentioned above.In addition,we also enumerate of complementary equienergetic strongly regular graphs that satisfy certain laws with other parameters,such as λ=tμ（2≤t≤5）,λ=t+μ（t∈Z⁺）,λ=μ-t（t∈Z⁺）and so on.In addition,we give some parameters of equienergetic strongly regular graphs and cone graphs with three eigenvalues.Finally,we combine the feature of the spectra of the graph with four different eigenvalues to construct equienergetic graphs with four different eigenvalues on the basis of strongly regular graph.Firstly,we provide equienergetic （?） and （?）,and construct equienergetic line graphs:L（（?））and L（G）,L（G）and L(K_p,2),L（G）and L(K_p,3),L（G）and L(K_p,5)（where G is a strongly regular graph）.Then,we enumerate parameters of equienergetic strongly regular graphs and cone graphs with four different eigenvalues.