On The Resistance Distance And Kirchhoff Index Of A Linear Hexagonal(Cylinder) Chain

The resistance between two vertices i and j in a connnected graph G is denoted by the equivalent resistance between i and j in N,where N is a resistor networks obtained from G by replacing each edge to a unit resistor.The resistance distance of a graph G is one of the most important measurement of quantifying structural properties for G,and it has widespread utility in physics,engineering,mathematics,chemistry and computer sciences.A hexagonal chain is a 2-connected graph consisting with some regular hexagon of unit edge length,in which satisfying each vertex in it belongs to at most two hexagons and each hexagon in it has at most two adjacent hexagons.In a hexag-onal chain,if all hexagons in the chain has two 2-degree vertices except the initial and the final one,then the hexagonal chain is called linear hexagonal chain.The graph obtained from identifing the initial edge wtih the terminal edge is called linear hexagonal cylinder chain.In this thesis,by the equivalent circuits and the pseudo inverse of discrete Laplacian matrix,some results about the resistance distance of linear hexagonal(resp.cylinder)chain are obtained.The concrete content is in the following:·In Chapter 1,we introduce the background and significance of the research and show the results we obtained in this thesis and its importance and innovations.·In Chapter 2,we give some necessary notations and terminologies,including the definitions of linear hexagonal chain and linear hexagonal cylinder chain.·In Chapter 3,we first obtain the explicit formulae for the resistance distance between any two vertices in the linear hexagonal chain.Then we determine the largest and the smallest resistances in the linear hexagonal chain.The monotonicity and asymptotic property of the resistances in linear hexagonal chain are discussed.·In Chapter 4,we first obtain the explicit formulae for the resistance distance between any two vertices in the linear hexagonal cylinder chain.Then we de-termine the largest and the smallest resistances in the linear hexagonal cylinder chain.As well the monotonicity and asymptotic property of resistances in linear hexagonal cylinder chain are discussed.·In Chapter 5,we determine the formula of Kirchhoff index of linear hexagonal(resp.cylinder)chain by using the formulae of resistance distances and find the ratio of them approximate to(?).·In Chapter 6,we summarize the main results in this paper and give some prospects for further research in the future.