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A signed graph is a graph whose edges are labelled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance Laplacian matrices. We characterize singularity and calculate the rank of these matrices and find signed distance Laplacian spectra of some classes of unbalanced signed graphs. We derive most of these results by proving them more generally for weighted signed graphs. [ABSTRACT FROM AUTHOR]

In this article, we study the signed distance matrix of the product of signed graphs such as the Cartesian product and the lexicographic product in terms of the signed distance matrices of the factor graphs. Also, we discuss the distance spectra of some special classes of product of signed graphs. [ABSTRACT FROM AUTHOR]

Abstract: In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended p-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of the Kronecker matrix product and the eigenvalues and energy of the product in terms of those of the factor graphs. For the Cartesian product we characterize balance and compute expressions for the Laplacian eigenvalues and Laplacian energy. We give exact results for those signed planar, cylindrical and toroidal grids which are Cartesian products of signed paths and cycles. We also treat the eigenvalues and energy of the line graphs of signed graphs, and the Laplacian eigenvalues and Laplacian energy in the regular case, with application to the line graphs of signed grids that are Cartesian products and to the line graphs of all-positive and all-negative complete graphs. [Copyright &y& Elsevier]

A signed graph is an ordered pair Σ = (G,σ), where G = (V, E) is the underlying graph of Σ with a signature function σ : E → {1, -1}. In this article, we define the nth power of a signed graph and discuss some properties of these powers of signed graphs. As we can define two types of signed graphs as the power of a signed graph, necessary and sufficient conditions are given for an nth power of a signed graph to be unique. Also, we characterize balanced power signed graphs. [ABSTRACT FROM AUTHOR]

In this paper, by defining two types of signed detour distances and corresponding detour distance matrices, we introduce the notion of detour distance compatibility for signed graphs and later applying these concepts, we give yet another characterization of balance in signed graphs. Further, we discuss signed detour spectra of certain classes of unbalanced signed graphs. [ABSTRACT FROM AUTHOR]

[Display omitted] • Microwave-solvothermal methods were used to synthesize the Pd@NSC-WO 3. • Hydroquinone, catechol, and resorcinol were exposed to photodegradation. • Synergistic effect, rapid charge transfers and reduced e-/h+ pair recombination. • Significant photocatalytic activity was achieved within 60 min under UV light. • It exhibits excellent photocatalytic activity, recyclability, and good stability. In this work, we successfully developed palladium nanoparticles anchored on nitrogen, sulfur-doped tungsten trioxide (Pd@NSC-WO 3) by microwave-solvothermal (MW-ST) technique. Herein, we report Pd@NSC-WO 3 as a multifunctional photocatalyst for phenolic pollutants (PPs) including hydroquinone (HQ), resorcinol (RC), and catechol (CC). The synthesized photocatalysts were characterized using X-ray diffraction (XRD), Brunauer-Emmett-Teller (BET) surface area analysis, hydrogen temperature programmed reduction (H 2 -TPR), Fourier transform infra-red (FT-IR), field emission scanning electron microscopy (FE-SEM), field emission transmission electron microscopy (FE-TEM), selected area electron diffraction (SAED), energy dispersive X-ray analysis (EDS), ultraviolet–visible spectroscopy (UV–vis), X-ray photoelectron spectroscopy (XPS), and electron paramagnetic resonance (EPR) techniques. It exhibits an enhanced photocatalytic degradation efficiency were archived under both UV and visible light (VL). The interaction between Pd and WO 3 and the presence of heteroatoms like N and S, which promote the photodegradation of PPs in the direct pathway and provided excellent stability. The formed intermediate products during PPs photodegradation were identified by liquid chromatography–mass spectrometry (LC–MS) technique. The trapping experiments confirmed that •OH and •O 2 − were active radicals during the degradation process. More importantly, after being tested in five cycles, Pd@NSC-WO 3 showed a good photocatalytic performance due to its chemical stability. In practice, the excellent properties associated with the novel Pd@NSC-WO 3 indicated that they could be considered as a promising light-driven photocatalyst to remove the PPs contamination in water. [ABSTRACT FROM AUTHOR]