Iskanje po repozitoriju

Opis: In this paper, we introduce the concept of k-integral graphs. A graph Γ is called k-integral if the extension degree of the splitting field of the characteristic polynomial of Γ over rational field ℚ is equal to k. We prove that for any positive integers k and Δ, the set of all finite connected graphs with algebraic degree at most k and maximum degree at most Δ is finite. We study 2-integral Cayley graphs over finite groups G with respect to Cayley sets which are a union of conjugacy classes of G. Among other general results, we completely characterize all finite abelian groups having a connected 2-integral Cayley graph with valency 2, 3, 4 and 5. Furthermore, we classify the finite groups G that al Cayley graphs over G with bounded valency are 2-integral.
Ključne besede: Cayley graph, algebraic degree, characters of groups, integral eigenvalue
Objavljeno v RUP: 18.08.2026; Ogledov: 298; Prenosov: 4
Celotno besedilo (481,80 KB)

Opis: 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.
Ključne besede: nut graph, Cayley graph, vertex-transitive graph, circulant graph, graph spectrum, graph eigenvalue
Objavljeno v RUP: 23.03.2026; Ogledov: 1008; Prenosov: 8
Celotno besedilo (404,59 KB)

Opis: Let k be odd, and n an odd multiple of 3. Although this can also be deduced from known results, we provide a new proof that Ck ⋊ C₈ and (Cn × C₃) ⋊ C₈ do not have the Directed Cayley Isomorphism (DCI) property. When k is prime, Ck ⋊ C₈ had previously been proved to have the Cayley Isomorphism (CI) property. To the best of our knowledge, the groups Cp ⋊ C₈ (where p is an odd prime) are only the second known infinite family of groups that have the CI property but do not have the DCI property. This also provides a new proof of the result (which follows from known results but was not explicitly published) that no group with an element of order 8 has the DCI property. One piece of our proof is a new result that may prove to be of independent interest: we show that if a permutation group has a regular subgroup of index 2 then it must be 2-closed.
Ključne besede: CI property, DCI property, Cayley graphs, Cayley digraphs, 2-closed groups, 2-closure
Objavljeno v RUP: 03.11.2025; Ogledov: 817; Prenosov: 7
Celotno besedilo (344,18 KB)

Opis: Let G be a group with identity element 1, and let S be a subset of G \ {1}. The subset S is called minimal if ⟨S⟩ = G and there exists an element s ∈ S such that ⟨S \ {s, s−1}⟩ ≠ G. In this paper, we completely determine all directed strongly regular Cayley graphs Cay(G, S) for any generalized dicyclic group G, provided that S is a minimal subset of G.
Ključne besede: directed strongly regular graph, Cayley graph, generalized dicyclic group
Objavljeno v RUP: 21.10.2025; Ogledov: 1012; Prenosov: 13
Celotno besedilo (385,94 KB)

Opis: By a result of Babai, with finitely many exceptions, every group G admits a semi-regular poset representation with three orbits, that is, a poset P with automorphism group Aut(P) ≃ G such that the action of Aut(P) on the underlying set is free and with three orbits. Among finite groups, only the trivial group and ℤ_2 have a regular poset representation (i.e. semi-regular with one orbit), however many infinite groups admit such a representation. In this paper we study non-necessarily finite groups which have a regular representation or a semi-regular representation with two orbits. We prove that if G admits a Cayley graph which is locally the Cayley graph of a free group, then it has a semi-regular representation of height 1 with two orbits. In this case we will see that any extension of the integers by G admits a regular representation. Applications are given to finite simple groups, hyperbolic groups, random groups and indicable groups.
Ključne besede: automorphism group of posets, Cayley graph, Dehn presentation, simple groups, random groups
Objavljeno v RUP: 21.10.2025; Ogledov: 978; Prenosov: 8
Celotno besedilo (431,19 KB)