Some of the most complex networks are those that (i) have been engineered under selective pressure (either economic or evolutionary), and (ii) are capable of eliciting network-level behaviors. Some examples are nervous systems, ant colonies, electronic circuits and computer software. Here we provide evidence that many such selected, behavioral networks are similar in at least four respects. (1) Differentiation: Nodes of different types are used in a combinatorial fashion to build network structures through local connections, and networks accommodate more structure types via increasing the number of node types in the network (i.e., increasing differentiation), not via increasing the length of structures. (2) Behavior: Structures are themselves combined globally to implement behaviors, and networks accommodate a greater behavioral repertoire via increasing the number of lower-level behavior types (including structures), not via increasing the length of behaviors. (3) Connectivity: In order for structures in behavioral networks to combine with other structures within a fixed behavior length, the network must maintain an invariant network diameter, and this is accomplished via increasing network connectivity in larger networks. (4) Compartmentalization: Finally, for reasons of economical wiring, behavioral networks become increasingly parcellated. Special attention is given to nervous systems and computer software, but data from a variety of other behavioral selected networks are also provided, including ant colonies, electronic circuits, web sites and businesses. A general framework is introduced illuminating why behavioral selected networks share these four correlates. Because the four above features appear to apply to computer software as well as to biological networks, computer software provides a useful framework for comprehending the large-scale function and organization of biological networks. © 2005 Wiley

This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices receive the same color, coloring the edges of a graph so that no adjacent edges receive the same color, and coloring the edges of a graph so that neither adjacent edges nor edges at distance one receive the same color. We use the model where colors on vertices must be chosen from assigned lists and consider the minimum size of lists needed to guarantee the existence of a proper coloring. More precisely, a list assignment function L assigns to each vertex a list of colors. A proper L-coloring is a proper coloring such that each vertex receives a color from its list. A graph is k-list-colorable if it has an L-coloring for every list assignment L that assigns each vertex a list of size k. The list chromatic number χl(G) of a graph G is the minimum k such that G is k-list-colorable. We also call the list chromatic number the choice number of the graph. If a graph is k-list-colorable, we call it k-choosable. The elements of a graph are its vertices and edges. A proper total coloring of a graph is a coloring

Abstract. Some results relating to the road-coloring conjecture of Alder, Goodwyn, and Weiss, which give rise to an O(n 2) algorithm to determine whether or not a given edge-coloring of a graph is a road-coloring, are noted. Probabilistic analysis is then used to show that, if the outdegree of every edge in an n-vertex digraph is δ = ω(log n), a road-coloring for the graph exists. An equivalent re-statement of the conjecture is then given in terms of the cross-product of two graphs. Definitions Let G be an n-vertex digraph. V (G) will denote the vertex-set of G, and E(G) will denote the edge-set of G. G is strongly connected if for every pair of vertices v and w in V (G), there is a directed path from v to w. The outdegree of vertex v ∈ V (G), d + (v), is the number of edges originating at v. G is aperiodic if the set of lengths of simple directed cycles in G has gcd 1. (See [Br] or [BR] for a discussion of aperiodic digraphs.) History The road-coloring conjecture of Alder, Goodwyn, and Weiss [AGW] (hereinafter referred to as RCC) is a graph-theoretic characterization of a problem from ergodic theory. Let G be a strongly connected digraph such that d + (v) = 2 for every v ∈ V (G). (In this paper, loops and multiple edges are not permitted. Strictly speaking, this is not essential to the problem, but it is required if we are to follow the path laid out by O’Brien [O] in theorem 3 below.) Let χ: E(G) → {R, B} be an edge coloring of G such that for each v ∈ V (G), v has one red (R) edge and one blue (B) edge going out from it. χ is called a road-coloring of G. A string I ∈ {R, B} ∗ will be 1

this paper. We show how such graphs can be defined and parameterised by two simple parameters, and generated stochastically in a manner analogous to the Erdos-Renyi model where the probability of an edge existing is range (scale) dependent. The vertices are to be thought of as ordered in a possibly incomplete one dimensional lattice, so that all edges inherit a natural length scale or range, derived form the distance between the end vertices in the underlying lattice ordering

Di#use gliomas are brain tumors that invade the surrounding normal tissue by an aggressive di#usion process. This di#use invasive behavior a#ects the prognosis adversely, and renders radical treatment impossible. Current mathematical models to quantify and analyze a cancer tumor are not scalable due to their enormous complexity. We developed a scalable, graph theoretical model, based on the spatial relationship between the cells, to quantify the properties of the invasion. The graph theoretical model is used by a machine learning algorithm. The learning algorithm uses graph metrics to distinguish (1) gliomas from surrounding normal tissue, and (ii) gliomas from inflammation. We tested the algorithms on real data to validate the proposed approach.

Nondeterministic space bounded computation and its unambiguous version have been the focus of attention because of their signi cance in various contexts. In particular, nondeterministic logspace NL has been the the focus of much attention because NL contains many natural computational problems.