Spectrum Of The Signless 1-Laplace Operators On Bicyclic Graphs

As an important branch of modern mathematics,graph theory is playing more and more important roles in mathematics and other scientific fields.The Laplace operators on graphs is an important research field,and since the last century,the research of the Laplace operators on graphs had achieved much progress,many important achievements had been obtained.In recent years,with the rise of neural networks and other disciplines,it had achieved remarkable progress in the application of artificial intelligence,machine learning and other fields.In recent years,after Chang putting forward the concept of 1-Laplace operators on graphs,the research of the 1-Laplace operators on the graph began to attract attention and a series of important theoretical results had been obtained.In particular,they proposed that the eigenvalues of the spectrum of 1-Laplace operators can be expressed by energy functions.On this basis,we study the energy function of signless 1-Laplace operators on bicyclic graphs.At the same time,we note that the energy function can be expressed by the signless incidence matrix of the graph.So by using the generalized inverse,we got some conclusions.The main results are as follows:First,we get some values of the energy function of the ?-graph.Then through the classification discussion,we prove that there is always a generalized inverse B₁ of a weighted signless incidence matrix,which makes the 1-norm of B₁ is equal to the inverse of the minimum value of the corresponding energy function,and the minimum value of the energy function of ?-graph is also given.Secondly,we consider the value of the energy function of the ?-graph,by dividing ?-graph into two classes,we also prove that there is always a generalized inverse B₁ of a weighted signless incidence matrix,which makes the 1-norm of B₁ is equal to the inverse of the corresponding energy function.On this basis,we also obtain the minimum value of the energy function of ?-graph.