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In this paper our focus is on regular signed graphs with exactly 3 (distinct) eigenvalues. We establish certain basic results; for example, we show that they are walk-regular. We also give some constructions and determine all the signed graphs with 3 eigenvalues, under the constraint that they are either signed line graphs or have vertex degree 3. We also report our result of computer search on those with at most 10 vertices.

We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

For a graph G we consider the problem of the existence of a switching equivalent signed graph with Laplacian eigenvalues that are all main and the problem of determination of all switching equivalent signed graphs with this spectral property. Using a computer search we confirm that apart from K 2 every connected graph with at most 7 vertices switches to at least one signed graph with the required property. This fails to hold for exactly 22 connected graphs with 8 vertices. If G is a cograph without repeated eigenvalues, then we give an iterative solution for the latter problem and the complete solution in the particular case when G is a threshold graph. The first problem is resolved positively for a particular class of chain graphs. The obtained results are applicable in control theory for generating controllable signed graphs based on Laplacian dynamics. [ABSTRACT FROM AUTHOR]

The recently discovered Cu46Zr33.5Hf13.5Al7 (at.%) bulk metallic glass (BMG) presents the highest glass-forming ability (GFA) among all known copper-based alloys, with a record-breaking critical casting thickness (or diameter) of 28.5 mm. At present, much remains to be explored about this new BMG that holds exceptional promise for engineering applications. Here, we report our study on the crystallization behavior of this new BMG, using isochronal and isothermal differential scanning calorimetry (DSC), X-ray diffraction (XRD), and transmission electron microscopy (TEM). With the calorimetric data, we determine the apparent activation energy of crystallization, the Avrami exponent, and the lower branch of the isothermal time–temperature–transformation (TTT) diagram. With XRD and TEM, we identify primary and secondary crystal phases utilizing samples crystallized to different degrees within the calorimeter. We also estimate the number density, nucleation rate, and growth rate of the primary crystals through TEM image analysis. Our results reveal that the crystallization in this BMG has a high activation energy of ≈360 kJ/mole and that the primary crystallization of this BMG produces a high number density (≈1021 m−3 at 475 °C) of slowly growing (growth rate < 0.5 nm/s at 475 °C) Cu10(Zr,Hf)7 nanocrystals dispersed in the glassy matrix, while the second crystallization event further produces a new phase, Cu(Zr,Hf)2. The results help us to understand the GFA and thermal stability of this new BMG and provide important guidance for its future engineering applications, including its usage as a precursor to glass–crystal composite or bulk nanocrystalline structures. [ABSTRACT FROM AUTHOR]

If G is a graph with n vertices, A G is its adjacency matrix and b is a binary vector of length n, then the pair (A G , b) is said to be controllable (or G is said to be controllable for the vector b ) if A G has no eigenvector orthogonal to b . In particular, if b is the all-1 vector j , then we simply say that G is controllable. In this paper, we consider the controllability of non-complete extended p-sums (for short, NEPSes) of graphs. We establish some general results and then focus the attention to the controllability of paths and related NEPSes. Moreover, the controllability of Cartesian products and tensor products is also considered. Certain related results concerning signless Laplacian matrices and signed graphs are reported. [ABSTRACT FROM AUTHOR]

We prove that for two graph classes, threshold graphs and chain graphs, there exist intervals in which their distance matrix does not have any eigenvalues. We also compute the determinant of the distance matrix of both threshold and chain graphs in terms of their generating binary sequence. In this way we positively address a research problem posed in [D.P. Jacobs, V. Trevisan, F. C. Tura, Distance eigenvalue location in thresholds graphs, Proceedings of DGA, Manaus; 2013. p. 1–4]. [ABSTRACT FROM AUTHOR]

In this paper we focus on connected signed graphs of fixed number of vertices, positive edges and negative edges that maximize the largest eigenvalue (also called the index) of their adjacency matrix. In the first step we determine these signed graphs in the set of signed generalized theta graphs. Concerning the general case, we use the eigenvector techniques for getting some structural properties of resulting signed graphs. In particular, we prove that positive edges induce nested split subgraphs, while negative edges induce double nested signed subgraphs.We observe that our concept can be applied when considering balancedness of signed graphs (the property that is extensively studied in both mathematical and non-mathematical context). [ABSTRACT FROM AUTHOR]