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Near-bipartite Leonard pairs
No full textLet denote a field, and let denote a vector space over with finite positive dimension. A Leonard pair on is an ordered pair of diagonalizable -linear maps and that each act on an eigenbasis for the other in an irreducible tridiagonal fashion. Let denote a Leonard pair on . Let denote an eigenbasis for on which acts in an irreducible tridiagonal fashion. For , define an -linear map such that and if . The map is called the flat part of . The Leonard pair is bipartite whenever . The Leonard pair is said to be near-bipartite whenever the pair is a Leonard pair on . In this case, the Leonard pair is bipartite and called the bipartite contraction of . Let denote a bipartite Leonard pair on . By a near-bipartite expansion of , we mean a near-bipartite Leonard pair on with bipartite contraction . In the present paper, we have three goals. Assuming is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over ; (ii) for each near-bipartite Leonard pair over we describe its bipartite contraction; (iii) for each bipartite Leonard pair over we describe its near-bipartite expansions. Our classification (i) is summarized as follows. We identify two families of Leonard pairs, said to have Krawtchouk type and dual -Krawtchouk type. A Leonard pair of dual -Krawtchouk type is said to be reinforced whenever for . A Leonard pair is said to be essentially bipartite whenever the flat part of is a scalar multiple of the identity. Assuming is algebraically closed, we show that a Leonard pair over with is near-bipartite if and only if at least one of the following holds: (i) is essentially bipartite; (ii) has reinforced dual -Krawtchouk type; and (iii) has Krawtchouk type
Numerical range for weighted Moore-Penrose inverse of tensor
No full textThis article first introduces the notion of weighted singular value decomposition (WSVD) of a tensor via the Einstein product. The WSVD is then used to compute the weighted Moore-Penrose inverse of an arbitrary-order tensor. We then define the notions of weighted normal tensor for an even-order square tensor and weighted tensor norm. Finally, we apply these to study the theory of numerical range for the weighted Moore-Penrose inverse of an even-order square tensor and exploit its several properties. We also obtain a few new results in matrix setting
Token graphs of Cayley graphs as lifts
No full textThis paper describes a general method for representing -token graphs of Cayley graphs as lifts of voltage graphs. This allows us to construct line graphs of circulant graphs and Johnson graphs as lift graphs on cyclic groups. As an application of the method, we derive the spectra of the considered token graphs. The method can also be applied to dealing with other matrices, such as the Laplacian or the signless Laplacian, and to construct token digraphs of Cayley digraphs
Multiplicativity of permanents over matrix semirings
No full textIn this paper, we investigate the conditions for the multiplicativity of the permanent over a matrix semiring. We prove that if is either a commutative antiring or a commutative semiring where the set of all additively invertible elements coincides with the set of all nilpotents, then the permanent is multiplicative on the group of invertible matrices over if and only if . We then use this result to investigate the number of invertible matrices over with a specified permanent
Qualitative, statistical, and extreme properties of spectral indices of signable pseudo-invertible graphs
No full textIn this paper, we investigate the Moore-Penrose inversion of a simple connected graph. We analyze qualitative, statistical, and extreme properties of spectral indices of signable pseudo-invertible graphs. We introduce and analyze a wide class of signable pseudo-invertible simple connected graphs. It is a generalization of the classical concept of positively integrally invertible graphs due to Godsil. We present several constructions of signable pseudo-invertible graphs. We also discuss statistical properties of various spectral indices of the class of signable pseudo-invertible graphs.
The vertex connectivity and the third largest eigenvalue in regular (multi-)graphs
No full textLet be a simple graph or a multigraph. The vertex connectivity of is the minimum size of a vertex set such that is disconnected or has only one vertex. We denote by the third largest eigenvalue of the adjacency matrix of . In this paper, we present an upper bound for in a -regular (multi-)graph which guarantees that , which is based on the result of Abiad et al. [Spectral bounds for the connectivity of regular graphs with given order. Electron. J. Linear Algebra 34:428-443, 2018]. Furthermore, we improve the upper bound for in a -regular multigraph which assures that
Sid Chaplin: A Writer with a Cause
No full textThe aim of this article is to recover the radical subtext of both the life and work of Sid Chaplin by reasserting the essentially political dimensions of his writing. Chaplin devoted the whole of his career as a writer to documenting not only the decline of the coal mining industry in the north-east of Britain where he lived, but he also traced the impact this process had on the working-class communities that were dependent on the pits. In his two later novels set in the city of Newcastle, The Day of the Sardine (1961) and The Watchers and the Watched (1962), Chaplin went on to dramatize similarly troubled changes in urban working-class life in the 1950s and 60s. The article not only argues that it is this nexus of class, politics and literature that translates so convincingly into his Newcastle novels, it also claims that it is the fundamental radicalism of his own literary project that explains the problematic neglect of his work by both critics and readers
“The Tale is Soon Told”: Working-Class Storytelling in Sylvia Pankhurst’s “Thrift” and May Westoby’s “The Injustice of the King”
No full textFrom 1880 onwards British Socialists produced a substantial body of short fiction using working-class oral literary traditions to frame their narratives. Women remain underrepresented in scholarship on this aspect of literary studies in part because much of what they wrote has been lost to history or remains hidden. Retrieving Sylvia Pankhurst’s “Thrift” (The Woman’s Dreadnought, June 1914) and May Westoby’s “The Injustice of the King” (Justice, November 1910) from the archives, and republishing them for a contemporary audience, contributes to restorative projects by scholars seeking to broaden the scope of writing on Socialist women’s creative activism by expanding the existing body of available work to include unpublished or neglected fiction, plays and poetry. These short stories, originally published in British newspapers, are examples of Pankhurst and Westoby’s ability to appeal to a socially situated readership of working men and women by simulating oral literatures, and more specifically, forms typical of the parable, fairy tale, moral tale, and working-class life writing. These stories exemplify how Socialist writers brought non-fiction and fiction into dialogue to simplify complex ideas, humanize the cold constraints of politics, draw connections between the social and the political, and pay tribute to British working-class culture and traditions
Working-class Academics: Challenging Deficit Narratives Through Cultural Wealth
No full textWhen navigating higher education (HE), working-class academics (WCAs) encounter persistent socioeconomic, cultural, and personal barriers throughout their academic careers. This study, grounded in the theoretical frameworks of Pierre Bourdieu and Tara Yosso, sought to illuminate the cultural wealth of WCAs. Employing a mixed-methods approach over a five-year period, the research engaged with 244 WCAs from various institutions across the United Kingdom (UK). The findings challenged the dominant deficit narratives surrounding WCAs, revealing that WCAs actively cultivate and leverage a rich array of cultural assets, encompassing examples of aspirational, navigational, linguistic, familial, social, and resistant capital. Two overarching themes emerged from the data: the profound impact of shared lived experiences in empowering marginalised students, and the crucial role of WCAs as change agents. The study demonstrates how WCAs employ their backgrounds as pedagogical assets while advocating for structural reforms. These findings suggest institutions should formally recognise working-class cultural wealth through revised hiring criteria, targeted support programmes, and inclusive decision-making frameworks. This research advances understanding of how marginalised groups can transform rather than simply adapt to institutional cultures
