Abstract:
We define a graded multiplication on the vector space of essential paths on a graph $G$ (a tree) and show that it is associative. In most interesting applications, this tree is an ADE Dynkin diagram. The vector space of length preserving endomorphisms of essential paths has a grading obtained from the length of paths and possesses several interesting bialgebra structures. One of these, the Double Triangle Algebra (DTA) of A. Ocneanu, is a particular kind of quantum groupoid (a weak Hopf algebra) and was studied elsewhere; its coproduct gives a filtrated convolution product on the dual vector space. Another bialgebra structure is obtained by replacing this filtered convolution product by a graded associative product.It can be obtained from the former by projection on a subspace of maximal grade, but it is interesting to define it directly, without using the DTA. What is obtained is a weak bialgebra, not a weak Hopf algebra.

Abstract:
Given any simple biorientable graph it is shown that there exists a weak {*}-Hopf algebra constructed on the vector space of graded endomorphisms of essential paths on the graph. This construction is based on a direct sum decomposition of the space of paths into orthogonal subspaces one of which is the space of essential paths. Two simple examples are worked out with certain detail, the ADE graph $A_{3}$ and the affine graph $A_{[2]}$. For the first example the weak {*}-Hopf algebra coincides with the so called double triangle algebra. No use is made of Ocneanu's cell calculus.

Abstract:
In this article a new formulation of the Weyl C*-algebra, which has been invented by Fleischhack, in terms of C*-dynamical systems is presented. The quantum configuration variables are given by the holonomies along paths in a graph. Functions depending on these quantum variables form the analytic holonomy C*-algebra. Each classical flux variable is quantised as an element of a flux group associated to a certain surface set and a graph. The quantised spatial diffeomorphisms are elements of the group of bisections of a finite graph system. Then different actions of the flux group associated to surfaces and the group of bisections on the analytic holonomy C*-algebra are studied. The Weyl C*-algebra for surfaces is generated by unitary operators, which implements the group-valued quantum flux operators, and certain functions depending on holonomies along paths that satisfy canonical commutation relations. Furthermore there is a unique pure state on the commutative Weyl C*-algebra for surfaces, which is a path- or graph-diffeomorphism invariant.

Abstract:
The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail.

Abstract:
The notion of equitable coloring was introduced by Meyer in 1973. In this paper we obtain interesting results regarding the equitable chromatic number = for the total graph of complete bigraphs (,), the central graph of cycles () and the central graph of paths ().

Abstract:
For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions

Abstract:
A radio labeling of a graph is a function from the vertex set to the set of nonnegative integers such that , where and are diameter and distance between and in graph , respectively. The radio number of is the smallest number such that has radio labeling with . We investigate radio number for total graph of paths. 1. Introduction In a telecommunication system to design radio networks, the interference constraints between a pair of transmitters play a vital role. For the transmitters of radio network, we seek to assign channels such that the network fulfills all the interference constraints. The assignment of channels to the transmitters is popularly known as channel assignment problem which was introduced by Hale [1]. For radio network if we assume that the frequencies are uniformly distributed in the spectrum then the frequency span determines the bandwidth allocated for the assignment. In this case, the interference between two transmitters is closely related with the geographic location of the transmitters. Earlier designer of radio networks considered only the two-level interference, namely, major and minor. They classified a pair of transmitters as very close transmitters if the interference level between them is major and close transmitters if the interference level between them is minor. To solve the channel assignment problem, the interference graph is developed and assignment of channels converted into graph labeling (a graph labeling is an assignment of label to each vertex according to certain rule). In interference graph, the transmitters are represented by the vertices, and two vertices are joined by an edge if corresponding transmitters have the major interference while two transmitters have minor interference then corresponding vertices are at distance two, and there is no interference between transmitters if they are at distance three or beyond it. In other words, very close transmitters are represented by adjacent vertices, and close transmitters are represented by the vertices which are at distance two apart. In fact, Roberts [2] proposed that a pair of transmitters which has minor interference must receive different channels and a pair of transmitters which has major interference must receive channels that are at least two apart. Motivated through this problem Griggs and Yeh [3] introduced -labeling in which channels are related with the nonnegative integers. Definition 1. A distance two labeling (or -labeling) of a graph is a function from vertex set to the set of nonnegative integers such that the following conditions are

Abstract:
For simply laced $SU(3)$ graphs we offer a geometric understanding of the path creation and annihilation operators for $SU(3)$ in terms of creation and annihilation of sequences of three vertices forming triangular cells or collapsed triangular cells. We prove that the space of paths of a given length can be decomposed as a direct sum of orthogonal sub-spaces constructed by recurrent applications of the path creation operator on subspaces of essential paths of shorter length.

Abstract:
The first step in the analysis of a structure is to generate its configuration. Different means are available for this purpose. The use of graph products is an example of such tools. In this paper, the use of product graphs is extended for the formation of different types of structural models. Here weighted graphs are used as the generators and the connectivity properties of different models are expressed in terms of the properties of their generators through simple algebraic relationships. In this paper by using graph product concepts and spatial structured matrices, a new algebraic closed form is proposed for mathematical formulation and presentation of structures. For clarification some examples are included. 1. Introduction Data generation is the first step in the analysis of every structure. Configuration processing of large scale problems without automatic approaches can be erroneous and occasionally impossible. Formex configuration processing is one such a means introduced by Nooshin [1] and further developed by Nooshin et al. [2] and Nooshin and Disney [3]. Similar methods are developed based on set theory by Behravesh et al. [4]. Kaveh applied graph theory for this formation [5] (see also Kaveh et al. [6]). The use of product graphs in structural mechanics is suggested in [7, 8] and application of the corresponding concepts utilizing the directed and looped generators is due to Kaveh and Koohestani [9], weighted graph products by Kaveh and Nouri [10] and weighted triangular and circular graph products employed by Kaveh and Beheshti [11]. There are many other references in the field of data generation; however, most of them are prepared for specific classes of a problem. For example, many algorithms have been developed and successfully implemented on mesh or grid generation; a complete review of which may be found in a paper by Thacker [12] and in the books by Thompson et al. [13], Liseikin [14], and Topping et al. [15]. In this paper the configuration processing of regular structures is considered. A structure is called regular if it can be considered as the product of two or three subgraphs (generators) [16]. The weighted graph products developed in [10] and their application are extended. Weighted paths and cycles are considered as the generators, and it is shown that many such product graphs can algebraically be expressed by simple relationships and a new algebraic closed form proposed for mathematical formulation and presentation of structures. Once this is done, then the existing methods can be applied to eigensolution and analysis of such

Abstract:
In this paper, we propose a scattered-field formulation for modelling Kerr and Raman nonlinear dispersive media effectsusing the Transmission Line Matrix method with the symmetrical condensed node (SCN -TLM). This model is based onnovel voltage sources and the Auxiliary Differential Equation (ADE) technique.