Let c be a proper k-coloring of a graph G. Let $\pi =\{R_1,R_2,\dots ,R_k\}$ be the partition of $V(G)$ induced by c, where $R_i$ is the partition class receiving color i. The color code $c_{\pi }(v)$ of a vertex v of G is the ordered k-tuple $(d(v,R_1),d(v,R_2),\dots ,$ $d(v,R_k)),$ where $d(v,R_i)$ is the minimum distance from v to each other vertex $u\in R_i$ for $1\le i\le k$ . If all vertices of G have distinct color codes, then c is called a locating k-coloring of G. The locating-chromatic number of G, denoted by ${\chi }_L(G)$ , is the smallest k such that G admits a locating coloring with k colors. In this paper, we give a characterization of the locating chromatic number of powers of paths. In addition, we find sharp upper and lower bounds for the locating chromatic number of powers of cycles.

Let R be a commutative ring with unity. The main objective of this article is to study the relationships between PP-rings, generalized morphic rings and EM-rings. Although PP-rings are included in the later rings, the converse is not in general true. We put necessary and sufficient conditions to ensure the converse using idealization and polynomial rings

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.

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.

A ring $R$ is called EM-Hermite if for each $a,b\in R$, there exist $% a_{1},b_{1},d\in R$ such that $a=a_{1}d,b=b_{1}d$ and the ideal $% (a_{1},b_{1})$ is regular. We give several characterizations of EM-Hermite rings analogue to those for K-Hermite rings, for example, $R$ is an EM-Hermite ring if and only if any matrix in $M_{n,m}(R)$ can be written as a product of a lower triangular matrix and a regular $m\times m$ matrix. We relate EM-Hermite rings to Armendariz rings, rings with a.c. condition, rings with property A, EM-rings, generalized morphic rings, and PP-rings. We show that for an EM-Hermite ring, the polynomial ring and localizations are also EM-Hermite rings, and show that any regular row can be extended to regular matrix. We relate EM-Hermite rings to weakly semi-Steinitz rings, and characterize the case at which every finitely generated $R$-module with finite free resolution of length 1 is free.

Let $G=(V,E)$ be a graph. The $k^{th}-$ power of $G$ denoted by $G^{k}$ is the graph whose vertex set is $V$ and in which two vertices are adjacent if and only if their distance in $G$ is at most $k.$ A vertex coloring of $G$ is acyclic if each bichromatic subgraph is a forest. A star coloring of $G$ is an acyclic coloring in which each bichromatic subgraph is a star forest.The minimum number of colors such that $G$ admits an acyclic (star) coloring is called the acyclic (star) chromatic number of G, and denoted by $\chi_{a}(G)(\chi_{s}(G))$. In this paper we prove that for $n\geq k+1,$ $\chi_{s}(P_{n}^{k})=\min\{\lfloor\frac{k+n+1}{2}\rfloor,2k+1\}$ and $\chi_{a}(P_{n}^{k})=k+1.$ Further we show that for $n\geq(k+1)^{2}$, $%2k+1\leq\chi _{s}(C_{n}^{k})\leq2k+2$ and $k+2\leq\chi_{a}(C_{n}^{k})\leq k+3.$ Finally, we derive the formula $\chi_{a}(C_{n}^{k})=k+2$ for $% n\geq(k+1)^{3}.$