The Education Department is a core unit within the faculty, responsible for planning, organizing, and overseeing educational activities. It works closely with academic staff to design and update course curricula, coordinate class schedules, and enhance the overall quality of teaching. The department aims to provide a supportive environment for effective learning and the academic development of students. It also plays a key role in academic advising, addressing educational concerns, and organizing consultation sessions. By applying modern teaching methods and responding to current educational needs, the Education Department strives to improve the learning process and contribute to student success.
The Wiener index of a simple graph is defined as the sum of distances between all vertices of the graph. It is well known that the Wiener index of a tree can be obtained as an edge additive quantity where edge contributions are given as the product of the number of vertices closer to each of the two end points of each edge. Thus the distances between vertices are not used for computing the Wiener index of trees. In a similar manner we introduce new topological indices which yields the Wiener, hyper-Wiener, Schultz and modified Schultz indices as special cases for trees. One advantage of this method is that in computing Schultz and modified Schultz of trees we need not take in to account the distances between vertices.
Gutman and Zhou (Relations between Wiener, hyper-Wiener and Zagreb indices, Chemical Physics Letters 394 (2004) 93-95) obtained some bounds on Wiener and hyper-Wiener indices, in term of the first Zagreb index in molecular graphs with girth greater than four. We obtain new inequalities for Wiener and hyperWiener indices, in terms of the first and the second Zagreb indices and the number of hexagons in these graphs. These inequalities improve the bounds obtained by Gutman and Zhou and are the best possible bounds. Using these relations we obtain a bound on the second Zagreb index in terms of the first Zagreb index, for hexagon-free graphs.
In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring (R, Ai), the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R-modules; (2) M =⊕λχλ Rωλ where λ is an index set and R/Ann(ωλ) is a principal ideal ring for each λ χ λ; (3) Every prime ideal of H is a direct sum of at most |A| cyclic fi-modules where A is an index set and M =⊕λχλ Rωλ; and (4) Every prime ideal of R is a summand of a direct sum of cyclic R-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring (R, M) is a direct sum of (at most n) principal ideals, it suffices to test only the maximal ideal M.
