Assuming that G is a finite group and H subgroup of G, the graph known as relative coprime with respect to (denoted Γ_(G,H)) has vertices corresponding elements G. Two distinct x y are adjacent by an edge if only (|x|,|y|)=1 or belongs H. This paper will focus on finding general formula for some topological indices dihedral group. The study in theory offers valuable insights into structural pro...

Let A be the adjacency matrix of a graph Γ . The nullity of A (that is, the dimension of the nullspace of A), when viewed as a matrix over a field of prime characteristic p, is called the p-nullity of Γ . We present several families of arc-transitive graphs with arbitrarily large p-nullity. We also show that the p-nullity of a vertextransitive graph of order a power of p is zero, provided that ...

We denote by f(n, k) the number of positive integers m no,AC{1,2,...,n} with lAl>f(n,2) (if61n then f(n,2)= in), then the coprime graph induced by A contains a complete tripartite graph on 2 Lc,,,~~~;,,, J + 1 vertices where one of the classes ...

Coprime linear microphone arrays allow for narrower beams with fewer sensors. A coprime microphone array consists of two staggered uniform linear subarrays with M and N microphones, where M and N are coprime with each other. By applying spatial filtering to both subarrays and combining their outputs, M+N-1 microphones yield M⋅N directional bands. In this work, the coprime sampling theory is imp...

Let P1,P2, . . . ,Pn be graph properties, a graph G is said to be uniquely (P1,P2, . . . ,Pn)-partitionable if there is exactly one (unordered) partition {V1, V2, . . . , Vn} of V (G) such that G[Vi] ∈ Pi for i = 1, 2, . . . , n. We prove that for additive and induced-hereditary properties uniquely (P1,P2, . . . ,Pn)-partitionable graphs exist if and 32 I. Broere, J. Bucko and P. Mihók only if ...

A graph G with vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers 1, 2, . . . , |V | such that for every edge xy in E, the labels assigned to x and y are relatively prime or coprime. A graph is called prime if it has a prime labeling. In this paper, we show that generalized Petersen graphs P (n, 3) are not prime for odd n, prime for even n ≤ 100 ...

Let G be a finite group. A coprime commutator in is any element that can written as [x,y] for suitable x,y∈G such π(x)∩π(y)=∅. Here π(g) denotes the set of prime divisors order g∈G. An anti-coprime an [x,y], where π(x)=π(y). The main results paper are follows. If |xG|≤n whenever x commutator, then has nilpotent subgroup n-bounded index. every x∈G, H nilpotency class at most 4 [G:H] and |γ4(H)| ...

Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely coprime corings (coprime comodules). Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate their relatio...

In this paper we characterize groups according to the number of end vertices in associated coprime graphs. An upper bound on order group that depends is obtained. We also prove $$2-$$ are only whose graphs have odd vertices. Classifications with small given. give a complete answer [4, Question 3.7], where show $$\mathbb {Z}_4$$ and {Z}_2\times \mathbb {Z}_2$$ graph has exactly three