  In this thesis, we study problems at the interface of analysis and discrete
mathematics. We discuss analogues of well known Hardy-type inequalities and
Rearrangement inequalities on the lattice graphs $\mathbb{Z}^d$, with a
particular focus on behaviour of sharp constants and optimizers.In the first
half of the thesis, we analyse Hardy inequalities on $\mathbb{Z}^d$, first for
$d=1$ and then for $d \geq 3$. We prove a sharp weighted Hardy inequality on
integers with power weights of the form $n^\alpha$. This is done via two
different methods, namely super-solution and Fourier method. We also use
Fourier method to prove a weighted Hardy type inequality for higher order
operators. After discussing the one dimensional case, we study the Hardy
inequality in higher dimensions ($d \geq 3$). In particular, we compute the
asymptotic behaviour of the sharp constant in the discrete Hardy inequality, as
$d \rightarrow \infty$. This is done by converting the inequality into a
continuous Hardy-type inequality on a torus for functions having zero average.
These continuous inequalities are new and interesting in themselves.
  In the second half, we focus our attention on analogues of Rearrangement
inequalities on lattice graphs. We begin by analysing the situation in
dimension one. We define various notions of rearrangements and prove the
corresponding Polya-Szeg\H{o} inequality. These inequalities are also applied
to prove some weighted Hardy inequalities on integers. Finally, we study
Rearrangement inequalities (Polya-Szeg\H{o}) on general graphs, with a
particular focus on lattice graphs $\mathbb{Z}^d$, for $d \geq 2$. We develop a
framework to study these inequalities, using which we derive concrete results
in dimension two. In particular, these results develop connections between
Polya-Szeg\H{o} inequality and various isoperimetric inequalities on graphs.
