Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using ${\vartheta }_i$-th Caputo-like operator. Bifurcation, phase portraits and the computation of the maximum Lyapunov Exponent $\left(L{E}_{max}\right)$ are used to demonstrate their impact on the system’s dynamics. Furthermore, we employ the sample entropy approach (SampEn), $C_0$ complexity and the 0-1 test to quantify complexity and validate chaos in the incommensurate system. Studies indicate that the discrete memristive system with incommensurate fractional orders manifests diverse dynamical behaviors, including hidden chaos, symmetry, and asymmetry attractors, which are influenced by the incommensurate derivative values. Moreover, a 2D non-linear controller is presented to stabilize and synchronize the novel system. The work results are provided by numerical simulation obtained using MATLAB R2024a codes.

According to recent research, discrete-time fractional-order models have greater potential to investigate behaviors, and chaotic maps with fractional derivative values exhibit rich dynamics. This manuscript studies the dynamics of a new fractional chaotic map-based three functions. We analyze the behaviors in commensurate and incommensurate orders, revealing their impact on dynamics. Through the maximum Lyapunov exponent $(L{E}_{max})$, phase portraits, and bifurcation charts. In addition, we assess the complexity and confirm the chaotic features in the map using the approximation entropy $ApEn$ and $C_0$ complexity. Studies show that the commensurate and incommensurate derivative values influence the fractional chaotic map-based three functions, which exhibit a variety of dynamical behaviors, such as hidden attractors, asymmetry, and symmetry. Moreover, the new system’s stabilizing employing a 3D nonlinear controller is introduced. Finally, our study validates the research results using the simulation MATLAB R2024a.

The traditional adjacency matrix of a mixed graph is not symmetric in general, hence its eigenvalues may be not real. To overcome this obstacle, several authors have recently defined and studied various Hermitian adjacency matrices of digraphs or mixed graphs. In this work we unify previous work and offer a new perspective on the subject by introducing the concept of monographs. Moreover, we consider questions of cospectrality.

Given the adjacency matrix A of a digraph, the eigenvalues of the matrix AA^T constitute the so-called non-negative spectrum of this digraph. We investigate the relation between the structure of digraphs and their non-negative spectra and associated eigenvectors. In particular, it turns out that the non-negative spectrum of a digraph can be derived from the traditional (adjacency) spectrum of certain undirected bipartite graphs.

In the theory of line graphs of undirected graphs, there exists an important theorem linking the incidence matrix of the root graph to the adjacency matrix of its line graph. For directed or mixed graphs, however, there exists no analogous result. The goal of this article is to present aligned definitions of the adjacency matrix, the incidence matrix, and line graph of a mixed graph such that the mentioned theorem is valid for mixed graphs.