Let F be an r-uniform hypergraph and G be a multigraph. The hypergraph F is a Berge-G if there is a bijection f : E(G) → E(F) such that e ⊆ f(e) for each e ∈ E(G). Given a family of multigraphs G, a hypergraph H is said to be G-free if for each G ∈ G, H does not contain a subhypergraph that is isomorphic to a Berge-G. We prove bounds on the maximum number of edges in an r-uniform linear hypergr...

The modified bubble-sort graph of dimension n is the Cayley graph of Sn generated by n cyclically adjacent transpositions. In the present paper, it is shown that the automorphism group of the modified bubble sort graph of dimension n is Sn×D2n, for all n ≥ 5. Thus, a complete structural description of the automorphism group of the modified bubble-sort graph is obtained. A similar direct product...

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist those automorphisms which leave invariant. We define Aut(G;C)-invariant normal subgroup Φ(G;C) has the property that, for any generators if we remove from it all elements Φ(G;C), then remaining is still G. The contains Frattini Φ(G) but inclusion may be proper. Cayley graph Cay(G,C) edge-transitive acts trans...

A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn . We characterize both classes of graphs in terms of a special class of Cayley graphs called normal Cayley graphs. This allows us to construct, for any integer n, a Cayley graph G such that Gn+1 → Gn 6→ Gn−1, answering a question of ...

A Cayley (di)graph Cay(G,S) of a group G with respect to set S ⊆ is said be normal if the image under its right regular representation in automorphism Cay(G,S), and called CI-(di)graph for every T Cay(G,S)≅ Cay(G,T), there α∈ Aut(G) such that Sα = T. finite DCI-group or an NDCI-group all digraphs are CI-digraphs, respectively, CI-group NCI-group graphs CI-graphs, respectively. Motivated by conj...

We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (subgroupoid), and only counts transitions between colour classes (cosets). These groupoids are employed towards a generic construction method for finite hypergraphs that realise specified...

A mixed graph is said to be second kind hermitian integral (HS-integral) if the eigenvalues of its Hermitian-adjacency matrix are integers. called Eisenstein (0, 1)-adjacency We characterize set S for which normal Cayley Cay(Γ,S) HS-integral any finite group Γ. further show that a and only it integral. This paper generalizes results Kadyan Bhattacharjy (2022) [11].