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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 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]

A signed graph Σ is an ordered pair (G , σ) , where G = (V , E) is the underlying graph of Σ with a signature function σ : E → { 1 , − 1 }. Notions of signed distance and distance-compatible signed graphs were introduced recently. We characterize distance compatibility in the case of a connected signed graph. Also, we provide characterizations for distance compatibility in the case of the Cartesian product, two lexicographic products and the tensor product of 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]

Green synthesis of bentonite-iron nanoparticles (Be-FeNPs) has been carried out using the leaf extract of Helicteres isora and its antibacterial and adsorption capacity for the removal of Methyl Blue (MB) dye have been also studied. The prepared Be-FeNPs and iron nanoparticles (FeNPs) are characterized by UV-visible, FTIR spectroscopic techniques and SEM analysis. The antibacterial activity of Be-FeNPs show good zone of inhibition against E.coli and S.aureus. Further the prepared composite (Be-FeNPs) and bentonite are used as adsorbents for the removal of Methyl Blue (MB) dye in aqueous solution under various experiment parameters like initial concentration, contact time and pH. The optimum initial pH, adsorbent concentration and time were determined to be 8.1, 45°C, and 100 ppm, 80 min for MB adsorption on to Be-FeNPs. The experimental data for MB adsorption fits well to the Freundlich and Langmuir isotherm models. The kinetic equations like Natarajan-Khalaf, Bhattacharya-Venkobachar and Lagergren equations were found to be applicable. The surface morphology of the Be-FeNPs before and after adsorption of dye revealed that it can be used as an excellent adsorbent for dye removal from dilute industrial effluents. [ABSTRACT FROM AUTHOR]

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]

Gain graphs are graphs where the edges are given some orientation and labeled with the elements (called gains) from a group so that gains are inverted when we reverse the direction of the edges. Generalizing the notion of gain graphs, skew gain graphs have the property that the gain of a reversed edge is the image of edge gain under an anti-involution. In this paper, we study two different types, Laplacian and g-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group Fx of a field F of characteristic zero. Defining incidence matrix, we also prove the matrix tree theorem for skew gain graphs in the case of the g-Laplacian matrix. [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]