Let $A = EJB + KL\in\mathbb R^{n\times n}$, where $E=\operatorname{diag}(1,0,0,...,0)$, $J$ is the all-one matrix, $B,K$ are positive diagonal matrices and $L$ is the Laplacian matrix of a connected undirected graph, i.e. $L$ is a positive semi-definite matrix with zero row/column sums and $L$ is irreducible (so that $L$ is not permutationally similar to a block diagonal matrix with proper diagonal sub-blocks).
I can show that $A$ is non-singular.
However, how can we derive a (sufficient) condition on the parameters $E$ and $K$ such that all the eigenvalues of $A$ have positive real part?
This is not true. Counterexample: \begin{aligned} A&=EJ\pmatrix{1\\ &1\\ &&120000\\ &&&120000} +\pmatrix{1\\ &10000\\ &&100\\ &&&100}\pmatrix{3&-1&-1&-1\\ -1&3&-1&-1\\ -1&-1&3&-1\\ -1&-1&-1&3}\\ &=\pmatrix{1&1&120000&120000\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0} +\pmatrix{3&-1&-1&-1\\ -10000&30000&-10000&-10000\\ -100&-100&300&-100\\ -100&-100&-100&300}\\ &=\pmatrix{4&0&119999&119999\\ -10000&30000&-10000&-10000\\ -100&-100&300&-100\\ -100&-100&-100&300}. \end{aligned} According to both Octave and WolframAlpha, $A$ has a conjugate pair of eigenvalues $-58.3941\pm5626.57i$ with negative real parts.