Manal Ghanem

In this paper, we work with mixed graphs, where some connections are undirected and others are directed. We study them using the Hermitian adjacency matrix and the Hermitian Laplacian. These matrices record both the underlying graph and the directions, and their eigenvalues change in a controlled way when we “switch” phases at vertices. A mixed graph is called balanced if a switching can remove all direction-phases. We measure how far a graph is from being balanced by frustration parameters: the minimum number of vertices or connections that must be removed to make the graph balanced. Using these parameters, we give explicit upper bounds on the smallest Hermitian Laplacian eigenvalue for connected unbalanced mixed graphs. We also give a bound that localizes this eigenvalue using a shortest frustrated cycle and its boundary. We then study the independence number. We prove a Hoffman-type upper bound that works without any regularity assumption by adding a variance term that measures how far the Hermitian row sums are from being constant. This variance changes under switching, so the bound can be improved by choosing a good switching. Paley-type examples show that the resulting bound can be much stronger than the inertia bound.

This article proposes a two-fold triple system (TFTS)-based obfuscation scheme as a structured physical-layer transmission technique for multiple-input multiple-output (MIMO) systems. TFTS is a class of combinatorial block designs that enables index-based spatial signaling without cryptographic keys or additional computational overhead. The proposed scheme constructs a space-modulation matrix derived from a TFTS of order v, where each row corresponds to a unique antenna activation pattern transmitting unmodulated carriers. The TFTS combinatorial space is generated by merging two non-isomorphic Steiner triple systems, yielding a mapping space that grows super-exponentially with v. This growth becomes extreme even for moderate dimensions; for example, at v = 40, the number of valid TFTS realizations exceeds 10¹⁰²⁵⁸, rendering exhaustive reconstruction computationally infeasible. The TFTS constellation matrix is assumed to be pre-shared and securely stored at the legitimate transceivers, while an eavesdropper is modeled as a fully informed and computationally unbounded adversary that lacks the correct TFTS mapping and therefore operates under mismatched decoding. Under this adversary model, Eve’s demodulation decisions degrade to random guesses, resulting in an average bit error rate (ABER) approaching 0.5 and negligible exploitable information. Accordingly, the proposed framework is explicitly positioned as a physical-layer obfuscation mechanism rather than a secrecy-capacity or cryptographic security scheme. The obfuscation strength of the system is evaluated through information-leakage analysis, quantifying the mutual information between transmitted indices and Eve’s observations, as well as detectability analysis based on hypothesis testing. Simulation results demonstrate that the proposed TFTS-obfuscation scheme preserves strong performance for the intended receiver in terms of ABER, mutual information, and achievable rate, while maintaining minimal information leakage and low detectability at the eavesdropper.

The \alpha-Hermitian adjacency matrix H_\alpha of a mixed graph X has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number \alpha. This enables us to define an incidence matrix of mixed graphs. Consequently, we define a generalization of line graphs as well as a generalization of the signless Laplacian adjacency matrix of graphs. We then study the spectral properties of the gamma-signless Laplacian adjacency matrix of a mixed graph. Lastly, we characterize when the signless Laplacian adjacency matrix of a mixed graph is singular and give lower and upper bounds of number of arcs and digons in terms of largest and lowest eigenvalue of the signless Laplacian adjacency matrix.