1. A note on Cayley nut graphs whose degree is divisible by fourIvan Damnjanović, 2026, original scientific article Abstract: A nut graph is a nontrivial simple graph such that its adjacency matrix has a one-dimensional null space spanned by a full vector. Fowler et al. in 2020 proved that there is a d-regular vertex-transitive nut graph of order n only if 4 ∣ d, 2 ∣ n, n ≥ d + 4 or d≡₄2, 4 ∣ n and n ≥ d + 6. It was recently shown that there exists a d-regular circulant nut graph of order n if and only if 4 ∣ d, 2 ∣ n, d > 0, together with n ≥ d + 4 if d≡₈4 and n ≥ d + 6 if 8 ∣ d, as well as (n, d) ≠ (16, 8) (in the paper from 2024). In this paper, we demonstrate the existence of a d-regular Cayley nut graph of order n for each n and d with 4 ∣ d, d > 0 and 2 ∣ n, n ≥ d + 4, thereby finding all the orders attainable by a Cayley nut graph, or vertex-transitive nut graph, with a fixed degree divisible by four.
Keywords: nut graph, Cayley graph, vertex-transitive graph, circulant graph, graph spectrum, graph eigenvalue
Published in RUP: 23.03.2026; Views: 155; Downloads: 4
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2. Role of music therapy in the development of language skills in children with autism spectrum disorder : a systematic literature reviewLucija Mlakar, Vesna Posavčević, 2026, review article Abstract: Autism Spectrum Disorder (ASD) is a complex neurodevelopmental condition that typically emerges in early childhood, marked by difficulties in communication, social interaction, behaviour, and emotional regulation. Despite these challenges, many children with ASD demonstrate exceptional musical abilities, making music a powerful medium for enhancing self-expression, fostering social bonds, and supporting neurological development crucial for speech and social skills. Historically, minimally verbal children with ASD were often excluded from research due to the difficulty of assessment using standardised tools; however, recent advancements have enabled more inclusive studies. Over the past decade, naturalistic approaches have gained prominence, with music therapy emerging as a particularly promising intervention. A systematic literature review, based on original research sourced from PubMed, Sage, and ScienceDirect, examined six studies involving children aged two to twelve years with minimal verbal abilities and a clinical diagnosis of autism. These studies consistently found that music therapy significantly supports the development of language and social communication skills, while also enhancing fronto-temporal brain connectivity. The review contributes valuable insights into the current state of research, underscores the importance of early intervention and parental involvement, and lays the groundwork for further exploration into the role of music therapy in language development for children with ASD.
Keywords: autism spectrum disorder, children, minimal language abilities, social communication, fronto-temporal brain connectivity, music therapy, non-music therapy
Published in RUP: 28.01.2026; Views: 476; Downloads: 13
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3. On the wreath product of signed and gain graphs and its spectrumMatteo Cavaleri, Alfredo Donno, Stefano Spessato, 2025, original scientific article Abstract: We introduce a notion of wreath product of two gain graphs (Γ_1, ψ_1, G_1) and (Γ_2, ψ_2, G_2), producing a gain graph over the direct product group G_2|V_Γ1| × G_1, whose underlying graph is the classical wreath product of graphs Γ_1≀Γ_2. By composition with a suitable group homomorphism, our construction produces a signed graph when the two factors are signed graphs. We prove that the wreath product is stable under switching isomorphism. By using group representations, we are able to perform spectral computations on the wreath product: in particular, we determine its largest and its smallest eigenvalue, and we give a description of the spectrum when the first factor is a complex unit complete balanced or antibalanced gain graph, and the second factor is circulant. Finally, when G_1 is a group of permutations of the vertex set of the first factor, and the group G_2 is abelian, we give an alternative definition producing a gain graph over the group wreath product G_1≀G_2, which turns out to be stable under switching equivalence of the second factor, when the first factor is balanced.
Keywords: gain graph, signed graph, wreath product of graphs, wreath product of groups, circulant gain graph, mixed Kronecker product, π-spectrum
Published in RUP: 22.10.2025; Views: 694; Downloads: 6
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4. Exploring the bounds and relationships of chemical graph indices : master's thesisArbër Avdullahu, 2023, master's thesis Keywords: energy of a graph, Randić index, first Zagreb index, IRB index, extremal graph, spectrum, eigenvalue, metaheuristic
Published in RUP: 05.10.2023; Views: 1935; Downloads: 20
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10. O ekstremnih grafih z dano stopnjo in premerom/ožino : doktorska disertacijaSlobodan Filipovski, 2018, doctoral dissertation Keywords: adjacency matrix, antipodal graphs, cages, excess, defect, Ramanujan graphs, selfrepeats, degree/diameter problem, spectrum, Moore graphs, asymptotic density, distance matrices, Bermond and Bollobas problem
Published in RUP: 21.01.2019; Views: 5651; Downloads: 0 |
