Does a balanced design have to be connected? Under a general block design setup, consider $v$ treatments allocated to $b$ blocks. Let $r_i$ and $k_j$ be the replication number of the $i$th treatment and $j$ the block respectively. We define $n_{ij}$ as the number of times the $i$th treatment appears in the $j$th block, $i=1,\ldots,v;j=1,\ldots,b$ and construct the incidence matrix $N=(n_{ij})_{v\times b}$. Let $R=\operatorname{diag}(r_1,r_2,\ldots,r_v)$ and $K=\operatorname{diag}(k_1,k_2,\ldots,k_b)$.
Then characteristic matrix of the block design corresponding to treatment effects is given by $C=R-NK^{-1}N'$.
Some relevant definitions and results from Theory of Block Designs by Aloke Dey:

A block design is connected if all elementary treatment contrasts are estimable.

Alternative definition due to R.C. Bose:

A treatment $i$ and a block $j$ in a block design are said to be associated if the treatment $i$ appears in the block $j$. Two treatments are said to be connected if it is possible to pass from one to the other through a chain consisting alternatively of treatments and blocks such that any two members of a chain are associated. A design is connected if every pair of treatments is connected.


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*A block design is connected if and only if $\operatorname{rank}(C)=v-1$.


A block design is (variance) balanced if it permits the estimation of all estimable normalized treatment contrasts with the same variance.


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*A connected block design is balanced if and only if all non-zero eigenvalues of $C$ are equal.


*A connected block design is balanced if and only if $C$ can be written as $C=(a-b)I_v+b\mathbf1_v\mathbf1_v'$ for some scalars $a,b$.
But how to verify if a design is variance-balanced when it is not connected? Is a balanced design always assumed to be connected?

Consider this worked out example from the book Experimental Designs: Exercises and Solutions by Gupta and Kabe:

They have stated that the theorem on eigenvalues of $C$ holds for any design (connected or not) and hence came to this conclusion. But this does not seem to complement the actual theorem. If a balanced design has to be connected, this answer cannot be right.
Can I answer whether the design in this example is balanced or not straight from definition?
 A: Summary:  Yes, a balanced block design have to be connected. 
You say you get different answers from different sources. If we are to discuss this, you must give references to those different sources with conflicting answers! Your reference is Theory of Block Designs by Aloke Dey. I do not have access to that, but to the newer INCOMPLETE BLOCK DESIGNS here by the same author, which seem to have been written as a modernized replacement. 
The reason for the answer is almost by definition: For a design to be balanced, first, all contrasts must be estimable, and that is basically the definition of connected. Below a short summary, with some definitions.
Notation is the same as used in the question. 
Definition 2.2.2 A block design is said to be connected if all treatment contrasts are estimable.  This is in fact equivalent to the graph theory definition used in Examples of connected designs in DOE.  This is discussed (among other things) in the very interesting paper at arXiv.
Definition of *estimable contrast (not given in that book, assumed) A contrast  $l^T\theta$ with $l\not= 0$ is estimable if an unbiased estimator of it exists. 
Lemma 2.2.4 A linear parametric function $p^T \tau$ (with $p\not= 0$) is estimable for a block design $d$ iff $p\in \mathcal{C}(C)$, that is, $p$ is in the column space of the designs C-matrix.
Theorem 2.2.1 A block design with $\nu$ treatments is connected iff (if and only if) $\DeclareMathOperator{\Rank}{Rank}   \Rank(C)=\nu-1$, where $C$ is its C-matrix or characteristic matrix.  
Definition 2.3.1 A connected block design is said to be variance-balanced if it permits the estimation of every normalized treatment contrast with the same variance.   So we can see that the conclusion is built into the definition, but even without that, the result is clear, since the requirement to be estimable with the same variance obviously presupposes to be estimable. 
Then there is the concept of efficiency balance, where much the same can be repeated.

Your example is $b=2$ blocks with $\nu=4$ treatments, block 1 with treatments 1,4 and block 2 with treatments 2,3. Write the treatment effects as $\tau_1,\tau_2,\tau_3, \tau_4$. Then, for example, the treatment contrast $\tau_2 -\tau_1$ is not estimable, so the design is not connected.  You can write out a linear model with a general mean, treatment effects and block effects, and see what happens.  We can also use lemma 2.2.4 above. $\tau_2-\tau_1 = p^T\tau$ with $p=(-1,1,0,0)^T$. The C-matrix of the design is 
$$
  C=\frac12\begin{pmatrix} 1 & 0& 0& -1 \\
                           0&1&-1&0\\
                           0&-1&1&0\\
                           -1&0&0&1
   \end{pmatrix}
$$  and it is clear that $p$ cannot be written as a linear combination of the columns of $C$.
But the intuitive idea of estimation of treatment contrasts in block designs, is to use really only within-block contrasts, and so average them. That is called the intra-block analysis. In the example, there is no blocks where both treatments 1 and 2 are used, so no within-block contrasts. 
