The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, the exact formulae for the Harary indices of tensor product G × Km0,m1,...,mr−1 and the strong product G⊠Km0,m1,...,mr−1 , whereKm0,m1,...,mr−1 is the complete multipartite graph with partite sets of sizes m0,m1, . . . ,mr−1 are obtained. Also upper bounds for th...

The notion of a symmetric group parametrized by elements is introduced. It shown that this an extension subgroup the wreath productG≀Sn</mml...

For a permutation group G acting on set V, subset I of is said to be an intersecting if for every pair elements g,h∈I there exists v∈V such that g(v)=h(v). The intersection density ρ(G) transitive the maximum value quotient |I|/|Gv| where Gv stabilizer point and runs over all sets in G. If largest then have Erdős–Ko–Rado (EKR)-property, moreover, has strict-EKR-property size coset stabilizer. I...

This paper is concerned with so-called index $d$ generalized cyclotomic mappings of a finite field $\mathbb{F}_q$, which are functions $\mathbb{F}_q\rightarrow\mathbb{F}_q$ that agree suitable monomial function $x\mapsto ax^r$ on each coset the subgroup $\mathbb{F}_q^{\ast}$. We discuss two important rewriting procedures in context and present applications thereof concern permutations $\mathbb{...

Definition 1.2 (Distinguishing number of a graph). Let Γ = (V,E) be a graph and let f : V → [r] be a coloring of the set of vertices by r colors. The map f need not be surjective, in fact when r > |V | it cannot be surjective. We say that f is r-distinguishing if the only automorphism of Γ that fixes the coloring f is the trivial automorphism. The distinguishing number of Γ is denoted by D(Γ) a...

In this paper, we show that the necessary and sufficient condition for the existence of an e Sk-factorization of (Cm ◦ Kn)∗ is n ≡ 0 (mod k(k − 1)), for all m > 3. In fact, our result together with a result of Ushio gives a complete solution for the existence of an e Sk-factorization of (Cm ◦Kn)∗ for all m ≥ 3. Further, we have obtained some necessary or sufficient conditions for the existence ...