Modern Graph Theory

Knots and Links: Preface

Let $\Omega $ be a connected graph and for a given $l$ -ordered partition of vertices of a connected graph $\Omega $ is represented as $\beta =\{\beta _{1},\beta _{2}, {\dots },\beta _{l}\}$ . The representation of a vertex $\mu \in V(\Omega)$ is the vector $r(\mu |\beta)=(d(\mu,\beta _{1}),d(\mu,\beta _{2}), {\dots }, d(\mu,\beta _{l}))$ . The partition $\beta $ is a resolving partition for vertices of $\Omega $ if all vertices of $\Omega $ having the unique representation with respect to $\beta $ . The minimum number of $l$ in the resolving partition for $\Omega $ is known as the partition dimension of $\Omega $ and represented as $pd(\Omega)$ . Resolving partition and partition dimension have multipurpose applications in networking, optimization, computer, mastermind games and modeling of chemical structures. The problem of computing constant values of partition dimension is NP-hard so one can find sharp bound for the partition dimension of graph. In this article, we computed the upper bound for the convex polytopes $\mathbb {E}_{n},~\mathbb {S}_{n},~\mathbb {T}_{n},~\mathbb {G}_{n},~\mathbb {Q}_{n}$ and flower graph $f_{n\times 3}$ .

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On Sharp Bounds on Partition Dimension of Convex Polytopes

Graph Theory with Applications to Engineering and Computer ... This outstanding introductory treatment of graph theory and its applications has had a long life in the instruction of advanced undergraduates and graduate students in all areas that require knowledge of this subject. The first nine chapters constitute an excellent overall introduction, requiring only some knowledge of set theory and matrix algebra.

Graph theory with applications to engineering and computer science, by Narsingh Deo. Pp xvii, 478. 1974. SBN 0 13 363473 6 (Prentice-Hall)

Vassiliev (finite type) invariants of knots can be described in terms of weight systems. These are functions on chord diagrams satisfying so-called 4-term relations. In the study of the sl2 weight system, it was shown that its value on a chord diagram depends on the intersection graph of the diagram rather than on the diagram itself. Moreover, it was shown that the value of this weight system on an intersection graph depends on the cy- cle matroid of the graph rather than on the graph itself. This result arose the question whether there is a natural way to introduce a 4-term relation on the space spanned by matroids, similar to the one for graphs. It happened however that the answer is negative: there are graphs having isomorphic cycle matroids such that applying the "second Vassiliev move" to a pair of corresponding vertices a;b of the graphs we obtain two graphs with nonisomorphic matroids. 
The goal of the present paper is to show that the situation is different for binary delta-matroids: one can define both the first and the second Vassiliev moves for binary delta-matroids and introduce a 4-term relation for them in such a way that the mapping taking a chord diagram to its delta-matroid respects the corresponding 4-term relations. Moreover, this mapping admits a natural extension to chord diagrams on several circles, which correspond to singular links. Delta-matroids were introduced by A. Bouchet for the purpose of studying embedded graphs, whence their relationship with (multiloop) chord diagrams is by no means unexpected. Some evidence for the existence of such a relationship can be found, for example, the Tutte polynomial for embedded graphs has been introduced. It was shown that this polynomial depends on the delta-matroid of the embedded graph rather than the graph itself and satises the Vassilev 4-term relation.

Delta-matroids and Vassiliev invariants

A classic exercise in the topology of surfaces is to show that, using handle slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a canonical form consisting of the connected sum of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Motivated by the principle that ribbon graph theory informs delta-matroid theory, we find the delta-matroid analogue of this surface classification. We show that, using a delta-matroid analogue of handle slides, every binary delta-matroid in which the empty set is feasible can be written in a canonical form consisting of the direct sum of the delta-matroids of orientable loops, and either non-orientable loops or pairs of interlaced orientable loops. Our delta-matroid results are compatible with the surface results in the sense that they are their ribbon graphic delta-matroidal analogues.

Handle slides for delta-matroids

We define a class of graphs called flower and give some properties of these graphs. Then the explicit expressions of the chromatic polynomial and the flow polynomial is given. Further, we give classes of graphs with the same chromatic and flow polynomials.

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Some Polynomials of Flower Graphs

We give explicit expressions of the Tutte polynomial of a complete flower graph and a flower graph with some petals missing.

http://bims.iranjournals.ir/article_38_af3b7b8c64bcf6a239e4239d09a47755.pdf

Tutte Polynomials of Flower Graphs

The 0-defect polynomial of a graph is just the chromatic polynomial. This polynomial has been widely studied in the literature. Yet little is known about the properties of k-defect polynomials of graphs in general, when 0 < k ≤  | E ( G )|. In this survey we give some properties of k -defect polynomials, in particular we highlight the properties of chromatic polynomials which also apply to k -defect polynomials. We discuss further research which can be done on the k -defect polynomials. Mathematics Subject Classification (2010): 05C15, 05C31. Keywords: Chromatic polynomial, k -defect polynomial, tutte polynomial, coboundary polynomial

An introduction to the k -defect polynomials

We nd an explicit expression of the Tutte polynomial of an n-fan. We also nd a formula of the Tutte polynomial of an n-wheel in terms of the Tutte polynomial of n-fans. Finally, we give an alternative expression of the Tutte polynomial of an n-wheel and then prove the explicit formula for the Tutte polynomial of an n-wheel.

http://bims.iranjournals.ir/article_449_8f1dce2d9821f65d9680e61e1825f866.pdf

Eunice Mphako-Banda

Papers

Handle slides for delta-matroids

Some Polynomials of Flower Graphs

Tutte Polynomials of Flower Graphs

An introduction to the k -defect polynomials

Tutte polynomials of wheels via generating functions