Star-like Graphs Research Articles

Evolutionary dynamics on small-order graphs

We study the stochastic birth-death model for structured finite populations popularized by Lieberman et al. [E. Lieberman, C. Hauert and M. A. Nowak (2005), Evolutionary dynamics on graphs, Nature, Vol. 433, pp. 312–316]. We consider all possible connected undirected graphs of orders three through eight. For each graph, using the Monte Carlo Markov Chain simulations, we determine the fixation probability of a mutant introduced at every possible vertex. We show that the fixation probability depends on the vertex and on the graph. A randomly placed mutant has the highest chances of fixation in a star graph, closely followed by star-like graphs, and the fixation probability is lowest for regular and almost regular graphs. We also find that within a fixed graph, the fixation probability of a mutant has a negative correlation with the degree of the starting vertex.

Maximizing the number of spanning trees in [formula omitted]-complements of asteroidal graphs

In this paper we introduce the class of graphs whose complements are asteroidal (star-like) graphs and derive closed formulas for the number of spanning trees of its members. The proposed results extend previous results for the classes of the multi-star and multi-complete/star graphs. Additionally, we prove maximization theorems that enable us to characterize the graphs whose complements are asteroidal graphs and possess a maximum number of spanning trees.

On representation of proteins by star-like graphs

To arrive at graphical representations of proteins one is confronted with number of arbitrary decisions how to assign the 20 natural amino acids to equivalent or non-equivalent sites of underlying geometrical objects used for construction of their graphical representation. Here we consider representation of proteins based on generalized star graphs, which are graphs with one vertex of maximal degree in the center to which are attached other vertices of either degree one or two. The matrix representation of proteins based on star-like graphs has an important advantage in that, while its pictorial representation depends on selected assignment of amino acids to various branches of star graph, its properties do not depend on the adopted assignment of vertices to amino acids. Hence, the derived graph invariants, devoid of artifacts associated with graphical representations of biosequences, will better reflect upon the inherent properties of protein structure. We describe several graph invariants, mostly extracted from distance matrices of star-like graphs, which can serve as protein descriptors. The approach is illustrated on strand A of the human insulin.

On the interval completion of chordal graphs

For a given graph G = ( V , E ) , the interval completion problem of G is to find an edge set F such that the supergraph H = ( V , E ∪ F ) of G is an interval graph and | F | is minimum. It has been shown that it is equivalent to the minimum sum cut problem, the profile minimization problem and a kind of graph searching problems. Furthermore, it has applications in computational biology, archaeology, and clone fingerprinting. In this paper, we show that it is NP-complete on split graphs and propose an efficient algorithm on primitive starlike graphs.

Edge clique graphs and some classes of chordal graphs

The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlike-threshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge clique graph is that the sizes of all maximal cliques and intersections of maximal cliques ought to be triangular numbers. We show that this condition is also sufficient for starlike-threshold graphs.

On the pathwidth of chordal graphs

In this paper we first show that the pathwidth problem for chordal graphs is NP-hard.Then we give polynomial algorithms for subclasses. One of those classes are the k-starlike graphs – a generalization of split graphs. The other class are the primitive starlike graphs – a class of graphs where the intersection behavior of maximal cliques is strongly restricted.