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Positive and negative square energies of graphs
No full textThe energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of . Let denote the sum of the squares of the positive and negative eigenvalues of , respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if is a connected graph of order , then and . In this paper, we show partial results towards this conjecture. In particular, numerous structural results that may help in proving the conjecture are derived, including the effect of various graph operations. These are then used to establish the conjecture for several graph classes, including graphs with certain fraction of positive eigenvalues and unicyclic graphs
Laughing all the Way to the Closed Factory: The Deindustrialization Comedy
No full textThis article explores the comedic treatment of deindustrialization in three films: Gung Ho (US, 1986), De frigjorte (Denmark, 1993), and The Full Monty (UK, 1997). Examining the films’ different ways of portraying deindustrialization, the article discusses how symptomatic it is that these films offered their audiences a form of comedic silver lining in an era when deindustrialization was still felt acutely as a crisis. Gung Ho’s comedic take on the 1980s deindustrialization crisis invokes hopeful discourses of reindustrialization, De frigjorte explores a crisis of masculinity after its protagonist is laid off after two decades’ employment at a local factory, and The Full Monty offers a story of men overcoming deindustrialization in a communal way. Reading these films in relation to each other, the article argues that these films offered viewers faced with the realities of deindustrialization a moment of comedic distance to economic hardship
#A11Y: EC Summer Institute on Accessibility
No full textIn this lesson, P-16+ administrators and educators were introduced to the rationale for creating accessible digital content, differences in terminology, and accessibility considerations through materials that were developed internally. Throughout the lesson, learners were presented with a variety of resources and were given the opportunity to apply the concepts. This two-day-long Summer Institute was designed for P-16+ individuals who are responsible for designing and/or procuring digital instructional content. The lesson was presented to K-12 educators throughout North Carolina who attended the 2022 NC Exceptional Children's Summer Institute
NEPS of complex unit gain graphs
No full textA complex unit gain graph (or -gain graph) is a gain graph with gains in , the multiplicative group of complex units. Extending a classical construction for simple graphs due to Cvektovic, suitably defined noncomplete extended -sums (NEPS, for short) of -gain graphs are considered in this paper. Structural properties of NEPS like balance and some spectral properties and invariants of their adjacency and Laplacian matrices are investigated, including the energy and the possible symmetry of the adjacency spectrum. It is also shown how NEPS are useful to obtain infinitely many integral graphs from the few at hands.Moreover, it is studied how NEPS of -gain graphs behave with respect to the property of being nut, i.e., having as simple adjacency eigenvalue and nowhere zero -eigenvectors. Finally, a family of new products generalizing NEPS is introduced, and their few first spectral properties explored
On eigenvalues of real symmetric interval matrices: Sharp bounds and disjointness
No full textIn this paper, the eigenvalue problem of real symmetric interval matrices is studied. First, in the case of real symmetric interval matrices, all the four endpoints of the two eigenvalue intervals are determined. These are not necessarily eigenvalues of vertex matrices, but it is shown that such a real symmetric interval matrix can be constructed from the original one. Then, necessary and sufficient conditions are provided for the disjointness of eigenvalue intervals. In the general case, due to Hertz, a set of special vertex matrices determines the maximal eigenvalue and a similar statement holds for the minimal one. In a special case, namely if the right endpoints of the off-diagonal intervals are not smaller than the absolute value of the left ones, he concluded the vertex matrix of the right endpoints provides the maximal eigenvalue. Generalizing it, it is shown that in the case of real symmetric interval matrices with special sign pattern, a single vertex matrix determines one of the extremal bounds
Multi-alternating sign matrices
No full textWe introduce a generalization of alternating sign matrices (ASMs) called multiASMs and develop some of their properties. Classes of multiASMs with specified row and column sum vectors and extend the classes of -matrices with specified and . The special case when is a constant vector, in particular all 2's, is treated in more detail. We also investigate the polytope spanned by a class of multiASMs. Finally, we discuss the possibility of defining a Bruhat order on a class of multiASMs
Relation between the row left rank of a quaternion unit gain graph and the rank of its underlying graph
No full textLet be a quaternion unit gain graph (or -gain graph), where is the underlying graph of , is the circle group, and is the gain function such that . Let be the adjacency matrix of and be the row left rank of . In this paper, we prove that , where and are the rank and the dimension of cycle space of , respectively. All corresponding extremal graphs are characterized. The results will generalize the corresponding results of signed graphs (Lu et al. [20] and Wang [33]), mixed graphs (Chen et al. [7]), and complex unit gain graphs (Lu et al. [21])
Diagonal-Schur complements of Nekrasov matrices
No full textThe Schur and diagonal-Schur complements are important tools in many fields. It was revealed that the diagonal-Schur complements of Nekrasov matrices with respect to the index set are Nekrasov matrices by Cvetkovic and Nedovic [Appl. Math. Comput., 208:225-230, 2009]. In this paper, we prove that the diagonal-Schur complements of Nekrasov matrices with respect to any index set are Nekrasov matrices. Similar results hold for -Nekrasov matrices. We also present some results on Nekrasov diagonally dominant degrees. Numerical examples are given to verify the correctness of the results
Middletown Lives through Middle-Class Eyes: Hillbilly Elegy and the Problem with the “Liberal Media”
No full textJ.D. Vance does not become Senator Vance without the success of Hillbilly Elegy, his best-selling memoir (and later, film) about growing up in, and getting out of, rural Appalachia. Initially praised by media critics for its ability to challenge middle-class assumptions about the “white working class,” the book assuaged both liberal anxiety and conservative outrage by providing demographically appropriate explanations for the election of Donald Trump. However, the book, feature film and subsequent political campaign are also part of a much larger, lucrative culture industry built upon the commodification and fetishization of the white working class, one driven by middle-class tastes and prejudices. This was most apparent in the promotion of the book and film by the so-called liberal media establishment, represented by the New York Times, The New Yorker, Netflix, Imagine Entertainment, HarperCollins, and Harpo Productions, to name a few. However, the reinforcement of the false binary between liberal and conservative media obscured how the corporate media system helped elect a candidate who will work most certainly against the interests of actual working people, further alienating them from each other and a shared labor platform more generally. Examining Hillbilly Elegy through the five filters of the Propaganda Model will help to explain the ideological and material effects of the corporate media’s agenda upon the growing class divide