The Kirchhoff Matrix Tree Theorem provides an efficient algorithm for determiningt(G), the number of spanning trees of any graphG, in terms of a determinant. However for many special classes of graphs, one can avoid the evaluation of a determinant, as there are simple, explicit formulas that give the value oft(G). In this work we show that many of these formulas can be simply derived from known properties of Chebyshev polynomials. This is demonstrated for wheels, fans, ladders, Moebius ladders, and squares of cycles. The method is then used to derive a new spanning tree formula for the complete prismR

 n
 (m) =K

 m
 ×C

 n
 . It is shown that
 
$$2^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\left( {1 - \frac{1}{{r - 1}} + o\left( 1 \right)} \right)} $$

 whereT

 n
 (x) is then
th order Chebyshev polynomial of the first kind.

Spanning tree formulas and chebyshev polynomials

A new analytical identification approach to the tooth contact points considering misalignments for spiral bevel or hypoid gears

In the social, behavioral, and economic sciences, it is important to predict which individual opinions eventually dominate in a large population, whether there will be a consensus, and how long it takes for a consensus to form. Such ideas have been studied heavily both in physics and in other disciplines, and the answers depend strongly both on how one models opinions and on the network structure on which opinions evolve. One model that was created to study consensus formation quantitatively is the Deffuant model, in which the opinion distribution of a population evolves via sequential random pairwise encounters. To consider heterogeneity of interactions in a population along with social influence, we study the Deffuant model on various network structures (deterministic synthetic networks, random synthetic networks, and social networks constructed from Facebook data). We numerically simulate the Deffuant model and conduct regression analyses to investigate the dependence of the time to reach steady states on various model parameters, including a confidence bound for opinion updates, the number of participating entities, and their willingness to compromise. We find that network structure and parameter values both have important effects on the convergence time and the number of steady-state opinion groups. For some network architectures, we observe that the relationship between the convergence time and model parameters undergoes a transition at a critical value of the confidence bound. For some networks, the steady-state opinion distribution also changes from consensus to multiple opinion groups at this critical value.

/pdf/opinion-formation-and-distribution-in-a-bounded-confidence-4hx68bhm9a.pdf

Opinion formation and distribution in a bounded-confidence model on various networks.

In this paper, we calculate the number of spanning trees in prism and antiprism graphs corresponding to the skeleton of a prism and an antiprism. By the electrically equivalent transformations and rules of weighted generating function, we obtain a relationship for the weighted number of spanning trees at the successive two generations. Using the knowledge of difference equations, we derive the analytical expressions for enumeration of spanning trees. In addition, we again calculate the number of spanning trees in Apollonian networks, which shows that this method is simple and effective. Finally we compare the entropy of our networks with other studied networks and find that the entropy of the antiprism graph is larger.

/pdf/counting-spanning-trees-in-prism-and-anti-prism-graphs-1gz7tuwhed.pdf

Counting spanning trees in prism and anti-prism graphs

Let Pn be a simple path on n vertices. An n-fan is a simple graph G formed from a path Pn by adding a vertex adjacent to every vertex of Pn. In this work we denote n-fan by Fn+1 and derive the explicit formula for t(Fn+1) the number of spanning trees in Fn+1 to be t(Fn+1 )=2 ((3− √ 5)/2) n+1 −((3+ √ 5)/2) n�1 5−3 √ 5 . In addition, we show that

/pdf/formulas-for-the-number-of-spanning-trees-in-a-fan-4g8i7p3wcw.pdf

Formulas for the Number of Spanning Trees in a Fan

Let the vertices of two disjoint, equal length cycles be labelled in one cycle and in the other. The prism Pn is defined as the graph obtained by adding to the disjoint cycles all edges of the form ViWi The prism is sometimes denoted by K 2×Cn . In this work we derive the following simple formula for t(Pn ) the number of spanning trees in .

F.T. Boesch

Papers

The number of spanning trees in a prism