In my research in graph theory i am getting symmetric matrices with trace zero of this kind
$$ \begin{bmatrix} 0 & 1 & 2 & 3 & 3 \\ 1 & 0 & 1 & 2 & 2 \\2 & 1 & 0 & 1 & 1 \\ 3 & 2 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 & 0 \end{bmatrix}$$ In general $$ D_{M} (G) = \begin{bmatrix} 0_{k\times k} & 1_{k\times k_{1}} & 2_{k\times k_{2}} & . . . & d_{k\times k_{d}}\\ 1_{k_{1}\times k} & 0_{k_{1}\times k_{1}} & 1_{k_{1}\times k_{2}} & . . . & (d-1)_{k_{1}\times k_{d}} \\ 2_{k_{2}\times k} & 1_{k_{2}\times k_{1}} & 0_{k_{2}\times k_{2}} & . . . & (d-2)_{k_{2}\times k_{d}}\\ . & . & . & . . . & .\\ . & . & . & . . . & .\\ . & . & . & . . . & .\\ d_{k_{d}\times k} & (d-1)_{k_{d}\times k_{1}} & (d-2)_{k_{d}\times k_{2}} & . . . & 0_{k_{d}\times k_{d}} \end{bmatrix}$$ Here the entries are all non-negative integers and $i_{m\times n}$ denotes a block whose entries are all i's 1. I find the matrices to be singular, can that be proved generally? 2. Also the numerical value of the largest eigenvalue increases with the increase in the value of even one entry, how to prove this?
For your first question, what you describe does not always hold. The easiest counterexample is $\det\pmatrix{0&1\\ 1&0}\ne0$. However, if $k>1$ or some $k_i>1$, the matrix is obviously singular because it has two identical rows.
As for your second question, if the matrix is zero, the assertion is obvious; if $d\ge1$, the assertion is still true because your matrix is entrywise nonnegative and irreducible (the matrix is primitive when $d\ge2$ and its entries are flipping between $0$ and $1$ when $d=1$). See this thread for detailed proofs.