In this study, we explore the concept of equilibrium measures within the framework of Schro & uml;dinger random walks on networks. Building on previous work, we demonstrate how these equilibrium measures can be leveraged to compute key network parameters, such as the mean first passage time and Kemeny's constant. By expressing these parameters in terms of generalized inverses of the associated M-matrix, we aim to provide new insights and efficient computational tools for network analysis, as demonstrated in the subsequent sections. The results are particularly applicable to both star and path networks, where we offer explicit formulations for these fundamental quantities. This work highlights the potential significance of equilibrium measures as a powerful tool for the study of complex networks.
The Drury permanent conjecture, published by Zhang in 2016 and disproved over the complex numbers by Hutchinson in 2017, has remained open in the real case, where a proof would imply the real case of the Chollet conjecture. The real 5 × 5 counterexample reported by Paksoy in 2023 concerns a different inequality, whose principal submatrices have order n −1 rather than order n−2 as in the conjecture; it is verified that Paksoy’s matrix satisfies the conjectured inequality strictly, under every simultaneous permutation of its rows and columns. The real case is then settled in the negative: an 8 × 8 real positive semidefinite matrix with integer entries and rank 3 is exhibited that violates the inequality, with an exact integer certificate of the violation. Direct sums with identity matrices give counterexamples of every order n ≥ 8.
It was recently proved that if a map defined on the ring of all $n\times n$ matrices satisfies the condition $f(x+y)=f(x)+f(y)$ for all invertible $x$ and $y$, then $f$ is additive. In the present paper, it is proved that the same result holds for the ring of $\mathbb{N}\times\mathbb{N}$ matrices with a finite number of nonzero entries in each column defined over a field of at least three elements.
A functional version of the block positivity characterization of normal matrices is established, thereby answering in a corrected form, a question posed in \cite{zhang2025more}. In particular, if \(A\in M_n\) and \(\alpha\in\mathbb{R}\setminus\{1\}\), we show that\[\begin{bmatrix}(A^*A)^{\alpha} & A^*\\A & (A^*A)^{1-\alpha}\end{bmatrix},\]is positive if and only if \(A\) is normal. An operator analog of this power-type criterion on infinite-dimensional Hilbert spaces is also proved; in this setting, the corresponding conclusion is formulated in terms of semi-hyponormality.
The spectral radius of a graph is a fundamental parameter closelylinked to network dynamics and robustness. Understanding its precisevariation under structural perturbations such as vertex deletion,edge removal, or graph coalescence is a central problem in spectralgraph theory. In this paper, we establish several sharp upper boundsthat quantify these changes with an explicit dependence on localgraph structures. For the deletion of an arbitrary nonempty propervertex set $S$, we give an exact Schur-complement formula for$\rho(G)$ in terms of $G[S]$, $G-S$, and the incidence matrix between them.The formula gives a precise relation between the spectral radius before and after deleting $S$ and yields a sharp upper bound withan equality characterization. In the special case where $S$ is a clique, this gives an explicit clique-deletion estimate. We also derive an alternative local bound for graph coalescence that captures the degrees at the identified root and is sometimes sharperthan the bound of Wang and Guo [(2020) \textit{Linear Algebra Appl.}, 601:101-112, 2020.]. For the deletion of the endpoints of an edge, we obtain a bound that incorporates the number of triangles containing the specified edge. We characterize equality in these upper bounds.
