This thesis comprises two main parts. The details of the two parts are as follows:

The first part of the thesis deals with the monopole-dimer model. The
dimer (resp. monomer-dimer) model deals with weighted enumeration of
perfect matchings (resp. matchings). The monopole-dimer model is a
signed variant of the monomer-dimer model which has determinantal
structure. A more general model called the *loop-vertex model* has
also been defined for an oriented graph and the partition function in
this case can also be written as a determinant. However, this model
depends on the orientation of the graph. The monopole-dimer model
interprets the loop-vertex model independent of the orientation for
planar graphs with Pfaffian orientation. The first part of the thesis
focuses on the extension of the monopole-dimer model for planar graphs
(Math. Phys. Anal. Geom., 2015) to Cartesian products thereof. We show
that the partition function of this model can be expressed as a
determinant of a generalised signed adjacency matrix. We then show that
the partition function is independent of the orientations of the planar
graphs so long as they are Pfaffian. When these planar graphs are
bipartite, we show that the computation of the partition function
becomes especially simple. We then give an explicit product formula for
the partition function of three-dimensional grid graphs a la Kasteleyn
and Temperley–Fischer, which turns out to be fourth power of a
polynomial when all grid lengths are even. Further, we generalise this
product formula to higher dimensions, again obtaining an explicit
product formula. We also discuss about the asymptotic formulas for the
free energy and monopole densities.

In 1999, Lu and Wu evaluated the partition function of the dimer model on two-dimensional grids embedded on a Möbius strip and a Klein bottle. We first prove a product formula for the partition function of the monopole-dimer model for the higher dimensional grid graphs with cylindrical and toroidal boundary conditions. We then consider the monopole-dimer model on high-dimensional Möbius and Klein grids, and evaluate the partition function for three-dimensional Möbius and Klein grids. Further, we show that the formula does not generalise for the higher dimensions in any natural way. Finally, we present a relation between the product formulas for three-dimensional grids with cylindrical and Möbius boundary conditions, generalising a result of Lu and Wu.

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph of $G$, denoted $Ec(G)$, is a graph defined on the vertices of $G$ in which two vertices are adjacent if one is eccentric to the other. In the second part of the thesis, we find the structure and the girth of the eccentric graph of trees, and see that the girth of the eccentric graph of a tree can either be zero, three, or four. Further, we study the structure of the eccentric graph of the Cartesian product of graphs and prove that the girth of the eccentric graph of the Cartesian product of trees can only be zero, three, four or six. Furthermore, we provide a complete classification of when the eccentric girth assumes these values. We also give the structure of the eccentric graph of the grid graphs and the Cartesian product of two cycles. Finally, we determine the conditions under which the eccentricity matrix of the Cartesian product of trees becomes invertible.

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Last updated: 15 Jul 2024