Esra Öztürk Sözen

In this paper, we introduce the Euler hyperbolic Sombor index and investigate its fundamental properties across various classes of graphs. We first establish general lower and upper bounds for this index and examine its relationships with several well-known topological indices, including the Euler Sombor index, the geometric-arithmetic index, the first Zagreb index, and the hyperbolic Sombor index. Furthermore, explicit formulas and extremal results are obtained for particular graph classes such as paths, cycles, stars, complete graphs, and trees, and the corresponding equality cases are characterized. In the application part, we present general combinatorial formulas for computing this index for benzenoid systems and phenylenes.

Institute of Mathematics of the Czech Academy of Sciences

In this essay (amply) cofinitely ?ss-supplemented modules are presented and fundamental algebraic features of these modules are examined. Privately, a ring characterization theorem is presented as follows. R is a ?ss-perfect ring if and only if every (projective) left R-module is (amply) cofinitely ?ss-supplemented. Moreover, the question when cofinitely ?ss-supplemented modules are cofinitely ss-supplemented is checked. With this aim we define left ?ss-rings and the fact that a ring R is a left ?ss-ring if and only if each cofinitely ?ss-supplemented R-module is cofinitely ss-supplemented is proven.

This article aims to explore the theoretical properties of topological descriptors used in fuzzy graph theory, which combines elements of graph theory and fuzzy set theory, within algebraic structures. For this purpose, fuzzy F-index (briefly F F-index) has been formulated theoretically for the fuzzy zero divisor graphs of the commutative ringn Z, where n = ??,?1?2, ?21 ?2, ?21 ?22, ?1 ?2 ?3 (?1, ?2, ?3 are primes and ? ? 3). In particular, a SageMath-based drawing algorithm that embodies the fuzzy graph structures of the rings is presented for application based convenience. Furthermore, a MATLbased code was created that directly calculate the fuzzy F-index of the fuzzy zero divisor graphs for every non-prime value of n > 1.

Graph theory is widely used to represent and analyze chemical structures. In addition, topological indices developed for graphs have a connection with the relationships of chemical structures such as physicochemical and bioactivity. Topological indices are widely used in QSPR-QSAR analysis and have found many applications in chemical graph theory. The oldest known degree-dependent topological indices are the first and second Zagreb indices. These indices have found wide application in chemical structures. Phenylenes containing aromatic and antiaromatic rings exhibit unique physicochemical properties and there is a wide variety of studies on phenylenes. In this article, we present some new formulas and lower bounds for the first and second Zagreb indices molecular structures of phenylenes. In addition, the BFS algorithm method, which is one of the graph algorithms, was used for the first time for the boundary study of chemical structures.

Background: Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Objective: Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. Methods: For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. Results: In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. Conclusion: In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.