Note on the game chromatic index of trees

We study edge coloring games defining the so-called game chromatic index of a graph. It has been reported that the game chromatic index of trees with maximum degree $\Delta = 3$ is at most $\Delta + 1$. We show that the same holds true in case $\Delta \geq 6$, which would leave only the cases $\Delta = 4$ and $\Delta = 5$ open. \u

Spectral analysis and a closest tree method for genetic sequences

We describe a new method for estimating the evolutionary tree linking a collection of species from their aligned four-state genetic sequences. This method, which can be adapted to provide a branch-and-bound algorithm, is statistically consistent provided the sequences have evolved according to a standard stochastic model of nucleotide mutation. Our approach exploits a recent group-theoretic description of this model

Not all phylogenetic networks are leaf-reconstructible

Unrooted phylogenetic networks are graphs used to represent reticulate evolutionary relationships. Accura

Are randomly grown graphs really random?

We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at delta = 1/8. At the transition, the average component size jumps discontinuously but remains finite. In contrast, a static random graph with the same degree distribution exhibits a second-order phase transition at delta = 1/4, and the average component size diverges there. These dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph--older vertices tend to have higher degree, and to link with other high-degree vertices, merely by virtue of their age. We conclude that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.Comment: 8 pages, 5 figure

World-Wide Web scaling exponent from Simon's 1955 model

Recently, statistical properties of the World-Wide Web have attracted considerable attention when self-similar regimes have been observed in the scaling of its link structure. Here we recall a classical model for general scaling phenomena and argue that it offers an explanation for the World-Wide Web's scaling exponent when combined with a recent measurement of internet growth.Comment: 1 page RevTeX, no figure

Quasistatic Scale-free Networks

A network is formed using the $N$ sites of an one-dimensional lattice in the shape of a ring as nodes and each node with the initial degree $k_{in}=2$. $N$ links are then introduced to this network, each link starts from a distinct node, the other end being connected to any other node with degree $k$ randomly selected with an attachment probability proportional to $k^{\alpha}$. Tuning the control parameter $\alpha$ we observe a transition where the average degree of the largest node $$ changes its variation from $N^0$ to $N$ at a specific transition point of $\alpha_c$. The network is scale-free i.e., the nodal degree distribution has a power law decay for $\alpha \ge \alpha_c$.Comment: 4 pages, 5 figure

Pseudofractal Scale-free Web

We find that scale-free random networks are excellently modeled by a deterministic graph. This graph has a discrete degree distribution (degree is the number of connections of a vertex) which is characterized by a power-law with exponent $\gamma=1+\ln3/\ln2$. Properties of this simple structure are surprisingly close to those of growing random scale-free networks with $\gamma$ in the most interesting region, between 2 and 3. We succeed to find exactly and numerically with high precision all main characteristics of the graph. In particular, we obtain the exact shortest-path-length distribution. For the large network ($\ln N \gg 1$) the distribution tends to a Gaussian of width $\sim \sqrt{\ln N}$ centered at $\bar{\ell} \sim \ln N$. We show that the eigenvalue spectrum of the adjacency matrix of the graph has a power-law tail with exponent $2+\gamma$.Comment: 5 pages, 3 figure

A Geometric Fractal Growth Model for Scale Free Networks

We introduce a deterministic model for scale-free networks, whose degree distribution follows a power-law with the exponent $\gamma$. At each time step, each vertex generates its offsprings, whose number is proportional to the degree of that vertex with proportionality constant m-1 (m>1). We consider the two cases: first, each offspring is connected to its parent vertex only, forming a tree structure, and secondly, it is connected to both its parent and grandparent vertices, forming a loop structure. We find that both models exhibit power-law behaviors in their degree distributions with the exponent $\gamma=1+\ln (2m-1)/\ln m$. Thus, by tuning m, the degree exponent can be adjusted in the range, $2 <\gamma < 3$. We also solve analytically a mean shortest-path distance d between two vertices for the tree structure, showing the small-world behavior, that is, $d\sim \ln N/\ln {\bar k}$, where N is system size, and $\bar k$ is the mean degree. Finally, we consider the case that the number of offsprings is the same for all vertices, and find that the degree distribution exhibits an exponential-decay behavior