Abstract:
The integrated management of multi-providers equipments is a key asset for telecommunication operators or service providers when selecting the appropriate network and service management platform fortheir network. In this paper, we present an open and adaptable platform that support fault and configuration management for next-generation networks. This platform Named Nagios based – information technology management system (NB-ITMS) leverage of the well-known Nagios platform to implement new capabilities. Offering additional features applications and management functions enable easy and low cost management of advanced services and networks technologies. The performance of our platform has been compared to existing off-the-shell platforms.

Abstract:
The integrated management of multi-providers equipments is a key asset for telecommunication operators or service providers when selecting the appropriate network and service management platform for their network. In this paper, we present an open and adaptable platform that support fault and configuration management for next-generation networks. This platform Named Nagios based - information technology management system (NB-ITMS) leverage of the well-known Nagios platform to implement new capabilities. Offering additional features applications and management functions enable easy and low cost management of advanced services and networks technologies. The performance of our platform has been compared to existing off-the-shell platforms.

Abstract:
Network management is one of the ways to decrease costs for network operation and loss decreasing, caused by network falls. Some tasks in network management are performed by monitoring system. The present systems of moni-toring of network with a dynamic topology are not suitable for networks with dynamic topology. Among free software, due to its functional characteristics Nagios can be singled out, which was designed to monitoring of static networks. However, Nagios can set network topology of different types including network with dynamic topology. Topology of networks with a dynamic topology is more changeable than in networks with static topology. Practical experience shows that Nagios is sensitive to peak demand on a hardware it is installed. In the current version of Nagios, information of network nodes must be entered manually. To avoid loss of infor-mation on network and nodes state, with topology change, it is required to deter-mine the number of temporary selected centers and isolated networks.

Abstract:
Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. Using the recognition principle of Fiedorowicz, we prove that spineless cacti are equivalent as operads to the little discs operad. It turns out that in terms of spineless cacti Cohen's Gerstenhaber structure and Fiedorowicz' braided operad structure are given by the same explicit chains. We also prove that spineless cacti and cacti are homotopy equivalent to their normalized versions as quasi-operads by showing that both types of cacti are semi-direct products of the quasi-operad of their normalized versions with a re-scaling operad based on R>0. Furthermore, we introduce the notion of bi-crossed products of quasi-operads and show that the cacti proper are a bi-crossed product of the operad of cacti without spines and the operad based on the monoid given by the circle group S^1. We also prove that this particular bi-crossed operad product is homotopy equivalent to the semi-direct product of the spineless cacti with the group S^1. This implies that cacti are equivalent to the framed little discs operad. These results lead to new CW models for the little discs and the framed little discs operad.

Abstract:
In this paper chain cacti are considered. First, for two specific classes of chain cacti (orto-chains and meta-chains of cycles with h vertices) the recurrence relation for independence polynomial is derived. That recurrence relation is then used in deriving explicit expressions for independence number and number of maximum independent sets for such chains. Also, the recurrence relation for total number of independent sets for such graphs is derived. Finaly, the proof is provided that orto-chains and meta-chains are the only extremal chain cacti with respect to total number of independent sets (orto-chains minimal and meta-chains maximal).

Abstract:
Motivated by the second author's construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed topological space $(Y,\bullet)$. These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra $C$. We show that the homology of the topological operad of based $Y$-cacti is the linear operad of based $H_*(Y)$-cacti. In addition, we show that for every coalgebra $C$ the operad of based $C$-cacti is Koszul. To prove the latter result, we use the criterion of Koszulness for operads due to the first author, utilising the notion of a filtered distributive law between two quadratic operads. We also present a new proof of that criterion which works over the ground field of arbitrary characteristic.

Abstract:
The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.

Abstract:
The concept of resistance distance was first proposed by Klein and Randi\'c. The Kirchhoff index $Kf(G)$ of a graph $G$ is the sum of resistance distance between all pairs of vertices in $G$. A connected graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. Let $Cat(n;t)$ be the set of connected cacti possessing $n$ vertices and $t$ cycles, where $0\leq t \leq \lfloor\frac{n-1}{2}\rfloor$. In this paper, the maximum kirchhoff index of cacti are characterized, as well as the corresponding extremal graph.

Abstract:
Explicit recurrences are derived for the matching polynomials of the basic types of hexagonal cacti, the linear cactus and the star cactus and also for an associated graph, called the hexagonal crown. Tables of the polynomials are given for each type of graph. Explicit formulae are then obtained for the number of defect-d matchings in the graphs, for various values of d. In particular, formulae are derived for the number of perfect matchings in all three types of graphs. Finally, results are given for the total number of matchings in the graphs.