The Kirchhoff Index And Entropy Of Total Independence Number About Generalized Ladder

For the graph G, V is vertices set of G and S（?）V. If the vertices of S are independent mutually, then S is called a independent set of V. A k-independent set of G is the independent set which has k vertices in the graph G. Denoted the number of all independent sets of G by σ（G）, namely, where i_k（G） is the number of k-independent set of G, i₀（G） = 1. The Kirchhoff index of G, denoted by , where r（v_i, v_j） is defined to be the resistance distance between vertices viand vj. Generalized ladder is obtained by the subdivision ladder, and it contents following two properties:（1）It hasn’t a vertex which simultaneously contains three polygons;（2）It hasn’t a polygon which adjacent to the two or more polygons. In this paper, according to the property of the Fibonacci number, the property of the Chebyshev polynomials, Laplace’s theorem, Transfer matrix and so on, we discuss the total independence number and the relationship of related entropy and boundary in generalized straight ladder and generalized zigzag ladder, and give the accurate computational result of the Kirchhoff Index of generalized straight ladders and generalized ring ladder.For specific expression:1. In the chapter two, we obtain computational formula of the total independence number of generalized straight ladder, and discuss the relationship of two kinds of entropies and boundary about total independence number.2. In the chapter three, we obtain computational formula of the total independence number of generalized zigzag ladder, and discuss the relationship of two kinds of entropies and boundary about total independence number.3. In the chapter four, we give accurate computational result of the Kirchhoff Index of generalized straight ladder.4. In the chapter five, we give accurate computational result of the Kirchhoff Index of generalized ring ladder.