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There are six different classes of endomorphisms for a graph. The sets of these endomorphisms always form a chain under the inclusion of sets. In order to study these different endomorphisms more systematically, Böttcher and Knauer proposed the concept of the endomorphism type of a graph in 1992. In this paper, we explore the six different classes of endomorphisms of graph P(3m+1,3). In particular, the endomorphism type of P(3m+1,3) is given.

There are six classes of endomorphisms for a graph. The sets of these endomorphisms form a chain under the inclusion of sets. In order to systematically study these endomorphisms, Böttcher and Knauer defined the concepts of the endomorphism spectrum and endomorphism type of a graph in 1992. In this paper, based on the property and structure of the endomorphism monoids of graphs, six classes of endomorphisms of double-edge fan graphs are described. In particular, we give the endomorphism spectra and endomorphism types of double-edge fan graphs.

A weak homomorphism from graph G to graph H is a mapping f:V(G)→V(H), where either f(x)=f(y) or f(x),f(y)∈E(H) hold for all x,y∈E(G). A rectangular grid graph is formed by taking the Cartesian product of two paths. Counting weak homomorphisms is a fundamental problem in graph theory. In this paper, we present formulas for calculating the number of weak homomorphisms from paths to rectangular grid graphs. This count directly corresponds to the number of partial walks with length m within the rectangular grid graph, offering a combinatorial solution to this enumeration problem.

The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.

The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.

We can define six classes of endomorphisms on a graph, and they always form a chain based on set inclusion. The concepts of endomorphism type and endomorphism spectrum were introduced by Böttcher and Knauer in 1992. They provided a systematic and organized approach to study endomorphisms of graphs. In this paper, we focus on the double vertex wheel graphs and provide a detailed description of their endomorphism spectra and endomorphism types.

Given a commutative ring R R with unity, and an endomorphism f f on R R , the graph G ( R , f ) G(R,f) assigned to R R is a simple graph with vertex set R R , and two distinct vertices r r and s s are adjacent if r f ( s ) = 0 rf(s)=0 or s f ( r ) = 0 sf(r)=0 . This graph generalizes Beck’s zero-divisor graph G ( R ) G(R) . This extension enables a deeper insight into both the algebraic structure and graph-theoretic properties. We explore the properties of G ( R , f ) G(R,f) , such as connectivity, completeness, and cycles. We also determine the values for the diameter, girth, independence number, clique number, chromatic number, and domination number, sometimes under algebraic conditions on R R or f f . The study also reveals how algebraic properties of R R and f f relate to the graph-theoretic properties of G ( R , f ) G(R,f) . Applications to the ring Z n Z_n are provided with illustrative examples. The ties between G ( R , f ) G(R,f) and Beck’s graph G ( R ) G(R) are also presented offering new methods to tackle problems in the study of zero-divisor graphs.

The aim of this article is to introduce the concept of centrally-extended Jordan epimorphisms and proving that if is a non-commutative prime ring (∗-ring) of characteristic not two, and is a CE-Jordan epimorphism such that [ ), ] ∈ ) ([ ), ] ∈ )) for all ∈ , then is an order in a central simple algebra of dimension at most 4 over its center or there is an element in the extended centroid of such that ) = ( ) = ) for all ∈

In this paper, we first prove that the set End(S) consisting of all linear transformations of restricted Lie supertriple systems is a restricted Lie supertriple system. Furthermore, a new restricted Lie supertriple system S̆ is constructed and study its some related properties. Then we introduce the S –endomorphisms of restricted Lie supertriple systems and study their related properties.

Gangyong Lee, S. Tariq Rizvi, and Cosmin S. Roman studied Dual Rickart modules. The main purpose of this paper is to develop the properties Dual Rickart modules. We prove that a module M and indecomposable N if M is N-dual Rickart module, then either Hom(M, N) = 0 or every non zero R-homomorphism from M to N is a epimorphism. And we give various characterizations of some kind of rings in term of dual Rickart modules.