The Study Of Steady-state Degree Distribution In Anti-BA-1 And Anti-LCD-1 Random Model And Hosoya Polynomials On Nanocones

This paper is divided into two parts. The first is about two models in complex networks, the two models are BA model and LCD model, they are models that adding some vertexes and edges randomly at every time step. the degree distribution is time-dependent degree distribution in time t. We define the degree sequence of the network Gt byd(t)={d1(t), d2(t),…, dk(t),…}. where di(t) is the proportional of vertexes with degree i over a large graph. And if limâ†'∞ d(t) exists, then the degree sequence d(t) is called the steady-state degree distribution of the network.From the two model, we will change the degree-preferential rule of BA model and LCD model, and we will define Anti-BA-1 model and Anti-LCD-1 model, finally gain the steady-state degree distribution of the Anti-BA-1 model and the Anti-LCD-1 model’s by strict calculation and proof.we gain that the steady-state degree distribution of Anti-BA-1 model and Anti-LCD-1 model are exponential distribution with a base 1/2, This distribution is common in reality.The second is about the combinatorial property in the network, Specifically stud-ied several kind of typical chemical figure of Hosoya polynomials on carbon nanocones, carbon nanocone is a graph with a polygon in the center and edges around the hexagon. Hosoya polynomial of G is a distance-based polynomial and is defined as:H(G, x)= ∑{u,v}(?)V(G) xdG(u,v), where V(G) is the set of all vertices of G and d(u, v) is the dis-tance between vertices u and v. This paper main content is the carbon nanocone that has a quadrangle, pentagon or heptagon in the center,(hexagonal carbon nanocone has been researched), called CNC4[n], CNC5[n] and CNC7[n], which n is the layer, seen in graph one, at last we obtain the Hosoya polynomial and some important indexes of this graph, such as Wiener index, Hyper-Wiener index, Tratch-Stankevitch-Zefirow index and so on.