In this study, we present an epidemic model that characterizes the behavior of a financial network of globally operating stock markets. Since the long time series have a global memory effect, we represent our model by using the fractional calculus. This model operates on a network, where vertices are the stock markets and edges are constructed by the correlation distances. Thereafter, we find an analytical solution to commensurate system and use the well-known differential transform method to obtain...

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S\left(G\right)$, the Samuel compactification, and $E\left(M\right(G\left)\right)$, the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of $G=S_{\infty }$, leading us to define and investigate several new types...

This paper surveys frequency assignment problems coming up in planning wireless communication services. It particularly focuses on cellular mobile phone systems such as GSM, a technology that revolutionizes communication. Traditional vertex coloring provides a conceptual framework for the mathematical modeling of many frequency planning problems. This basic form, however, needs various extensions to cover technical and organizational side constraints. Among these ramifications are T-coloring and...

This is an extended version of a talk given by the author at the conference “Algebra and Topology in Interaction” on the occasion of the 70th Anniversary of D.B. Fuchs at UC Davis in September 2009. It is a brief survey of an area originated around 1995 by I. Gelfand and the speaker.

A form-finding problem for tensegrity structures is studied; given an abstract graph, we show an algorithm to provide a necessary condition for it to be the underlying graph of a tensegrity in Rd (typically d=2,3) with vertices in general position. Furthermore, for a certain class of graphs our algorithm allows to obtain necessary and sufficient conditions on the relative position of the vertices in order to underlie a tensegrity, for what we propose both a geometric and a symbolic approach.

Starting from the famous Königsberg bridge problem which Euler described in 1736, we intend to show that some results obtained 180 years later by König are very close to Euler's discoveries.

To every graph (or digraph) A, there is an associated automorphism group Aut(A). Frucht’s theorem asserts the converse association; that for any finite group G there is a graph (or digraph) A for which Aut(A) ∼= G. A new operation on digraphs was introduced recently as an aid in solving certain questions regarding cancellation over the direct product of digraphs. Given a digraph A, its factorial A! is certain digraph whose vertex set is the permutations of V (A). The arc set E(A!) forms a group,...

Dado un grafo G = (X,E) con un solo vértice insaturado p, se estudia el problema de encontrar, para todo x ∈ X, un camino M-alternado par que una x con p. Se halla un algoritmo, y se plantea su aplicación cara a dar una variante del Algoritmo de Edmonds en la que no haya que contraer los pseudovértices.

For each vertex v in a graph G, let there be associated a subgraph $H_v$ of G. The vertex v is said to dominate $H_v$ as well as dominate each vertex and edge of $H_v$. A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number ${\gamma }_{FH}\left(G\right)$. A full dominating set of G of cardinality ${\gamma }_{FH}\left(G\right)$ is called a ${\gamma }_{FH}$-set of G. We study three types of full domination in graphs: full...

Let $G_1$ and $G_2$ be copies of a graph $G$, and let $f:V\left(G_1\right)\to V\left(G_2\right)$ be a function. Then a functigraph $C(G,f)=(V,E)$ is a generalization of a permutation graph, where $V=V\left(G_1\right)\cup V\left(G_2\right)$ and $E=E\left(G_1\right)\cup E\left(G_2\right)\cup \{uv:u\in V\left(G_1\right),v\in V\left(G_2\right),v=f\left(u\right)\}$. In this paper, we study colorability and planarity of functigraphs.

In this note we consider a discrete symmetric function f(x, y) where $$f(x,a)+f(y,b)⩾f(y,a)+f(x,b)foranyx⩾yanda⩾b,$$ associated with the degrees of adjacent vertices in a tree. The extremal trees with respect to the corresponding graph invariant, defined as $$\sum _{uv\in E\left(T\right)}f\left(deg\right(u),deg(v\left)\right),$$ are characterized by the “greedy tree” and “alternating greedy tree”. This is achieved through simple generalizations of previously used ideas on similar questions. As special cases, the already known extremal structures of the Randic index follow as corollaries. The extremal structures...