A connected acyclic graph in which the degree of every vertex is at most four is called a molecular tree. A number associated with a molecular tree that can help to approximate the physical or ...chemical properties of the corresponding molecule is called a topological index. It is of great importance to investigate the relation between the structure and the thermodynamic properties of those molecules. In this paper, we investigated the extreme value of the first reformulated Zagreb index with a given order and degree of a graph. Further, we investigated the molecular trees that attain the maximum and minimum values. As an application, we presented the regression models to predict the acentric factor and entropy of octane isomers. Our extremal graphs give the minimum and the maximum acentric factor and entropy, which satisfied the experimental values.
The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fqv〈v3−α22v〉 and Q=Fqu,v〈u2−α12,v3−α22v〉 are nonchain finite rings and αi is in ...Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.
Let $ \mathfrak{S} $ be a ring. The main objective of this paper is to analyze the structure of quotient rings, which are represented as $ \mathfrak{S}/\mathfrak{P} $, where $ \mathfrak{S} $ is an ...arbitrary ring and $ \mathfrak{P} $ is a prime ideal of $ \mathfrak{S} $. The paper aims to establish a link between the structure of these rings and the behaviour of traces of symmetric $ n $-derivations satisfying some algebraic identities involving prime ideals of an arbitrary ring $ \mathfrak{S} $. Moreover, as an application of the main result, we investigate the structure of the quotient ring $ \mathfrak{S}/\mathfrak{P} $ and traces of symmetric $ n $-derivations.
Let n≥2 be a fixed integer and A be a Csup.∗-algebra. A permuting n-linear map G:Asup.n→A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D:Asup.n→A such ...that G(ςsub.1,ςsub.2,…,ςsub.iςsub.i sup.′,…,ςsub.n)=G(ςsub.1,ςsub.2,…,ςsub.i,…,ςsub.n)ςsub.i sup.′+ςsub.iD(ςsub.1,ςsub.2,…,ςsub.i sup.′,…,ςsub.n) holds ∀ςsub.i,ςsub.i sup.′∈A. In this paper, we investigate the structure of Csup.∗-algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity.
Let A be a prime *-algebra. A product defined as U•V=UVsup.∗ +VUsup.∗ for any U,V∈A, is called a bi-skew Jordan product. A map ξ:A→A, defined as ξ(psub.n (Usub.1 ,Usub.2 ,⋯,Usub.n ))=∑sub.k=1 sup.n ...psub.n (Usub.1 ,Usub.2 ,...,Usub.k−1 ,ξ(Usub.k ),Usub.k+1 ,⋯,Usub.n ) for all Usub.1 ,Usub.2 ,...,Usub.n ∈A, is called a non-linear bi-skew Jordan n-derivation. In this article, it is shown that ξ is an additive ∗-derivation.
In this article, we investigate the formation of reversible cyclic codes (i.e., its codewords forms a symmetry) over the ring S=Fsub.2+uFsub.2+usup.2Fsub.2, where usup.3=0. We find a unique set of ...generators for cyclic codes over S and classify reversible cyclic codes to their generators. The dual reversible cyclic codes are studied as well. Moreover, we provide some examples of reversible cyclic codes.
The action of any group on itself by conjugation and the corresponding conjugacy relation plays an important role in group theory. Generalizing the group theoretic notion of conjugacy to semigroups ...is one of the interesting problems, and semigroup theorists had produced substantial amount of research in this direction. The challenge to introduce a new notion of conjugacy in semigroups is to choose the suitable set of conjugating elements. A semigroup may contain a zero, and if zero lies in the conjugating set, then the relation reduces to the universal relation as can be seen in the notions ∼ l , ∼ p , and ∼ o . To avoid this problem, various innovative notions of conjugacy in semigroups have been considered so far, and ∼ n is one of these notions. ∼ n is an equivalence relation in any semigroup, coincides with the usual group theoretic notion if the underlying semigroup is a group, and does not reduce to a universal relation even if S contains a zero. In this paper, we study ∼ n notion of conjugacy in some classes of epigroups. Since epigroups are generalizations of groups, our results of this paper are innovative and generalize the existing results on other notions. After proving some fundamental results, we compare our results with existing ones and prove that they are worth contribution in the study of conjugacy in epigroups.
Let R be a commutative ring with identity 1≠0 and let Z(R)′ be the set of all non-unit and non-zero elements of ring R. Γ′(R) denotes the cozero-divisor graph of R and is an undirected graph with ...vertex set Z(R)′ , w∉zR , and z∉wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs Γ′( Z n) . We also show that the cozero-divisor graph Γ′( Z p1p2) is a signless Laplacian integral.
Let R be a commutative ring with identity 1≠0 and let Z(R)sup.′ be the set of all non-unit and non-zero elements of ring R. Γsup.′ (R) denotes the cozero-divisor graph of R and is an undirected graph ...with vertex set Z(R)sup.′ , w∉zR, and z∉wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs Γsup.′ (Zsub.n ). We also show that the cozero-divisor graph Γsup.′ (Zsub.p1p2 ) is a signless Laplacian integral.
The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of Fsub.qPQ, where P=Fqv/〈v3−α22v〉 and Q=Fqu,v/〈u2−α12,v3−α22v〉 are nonchain finite rings and αsub.i ...is in Fsub.q/{0} for i∈{1,2}, where q=psup.m with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.
