iosr-Jm   Volume-21 ~ Issue-4 ~ Jul. – Aug. 2025

Abstract : In graphs, circular distance between two vertices is the sum of geodesic and detour distance between them. Based on circular distance, we introduce a novel vertex, termed as null vertex, that is distinct from interior and boundary vertices and discuss its presence in some graphs

Keywords: circular distance, boundary vertex, interior vertex, null vertex.

[1]. G. Chartrand and P. Zhang, A First Course in Graph Theory, Dover Publications, New York, 2012.
[2]. G. Chartrand, D. Erwin, G. Johns and P. Zhang, Boundary vertices in graphs, Discrete Mathematics, vol. 263, pp. 25–34, 2003. doi.org/10.1016/S0012-365X (02)00567-8
[3]. G. Chartrand, H. Escuadro and P. Zhang, Detour distance in graphs, J Combin. Math. Combin. Compute., vol. 53, pp. 75–94, 2005.
[4]. M.P. Sunil and J. Suresh Kumar, On Null Vertex in Fuzzy Graphs, International Journal of Analysis and Applications, vol. 22, 43, 2024. https://doi.org/10.28924/2291-8639-22-2024-43.
[5]. M.P. Sunil and J. Suresh Kumar, On Null Vertex in Bipolar Fuzzy Graphs, International Journal of Analysis and Applications, vol.22, 142, 2024. https://doi.org/10.28924/2291-8639-22-2024-142.

Abstract : This paper presents a novel algorithm for exact chromatic number computa- tion in graphs, with optimized applications to university course timetabling. We introduce a two-phase approach that: (1) leverages improved bound propagation techniques to reduce the search space, and (2) implements conflict-aware neigh- borhood operators for efficient feasible solution generation. Our method achieves a 23% reduction in average computation time compared to state-of-the-art exact methods on standard benchmarks (ITC-2007, Toronto), while maintaining 100% feasibility guarantees. Theoretical analysis proves the algorithm’s correct- ness, and empirical results....

KeywordsChromatic number, graph theory, NP-complete problem, university course assignment, planar graph, clique

[1]. Abdullah S, Taha H, Turabieh H. A Hybrid Variable Neighborhood Search For University Course Timetabling. Journal Of Scheduling. 2014;17(3):249–261.
[2]. Bettinelli A, Cacchiani V, Malaguti E. A Hybrid Metaheuristic Approach For The University Course Timetabling Problem. Journal Of Heuristics. 2015;21(1):1–23.
[3]. Br´Elaz D. New Methods To Color The Vertices Of A Graph. Communications Of The ACM. 1979;22(4):251–256.
[4]. Cambazard H, H´Ebrard E, O’Sullivan B, Papadopoulos A. Local Search And Con- Straint Programming For The Post Enrolment-Based Course Timetabling Problem. Annals Of Operations Research. 2010;194:111 – 135. Https://Api.Semanticscholar. Org/Corpusid:1538765.
[5]. Cambazard H, H´Ebrard E, O’Sullivan B, Papadopoulos A. Local Search And Constraint Programming For The Post-Enrolment Course Timetabling Problem. In: Proceed- Ings Of The 6th International Conference On Learning And Intelligent Optimization LION’ 2, Springer; 2012. P. 146–160..

Abstract : In this paper, the concepts of fuzzy neutrosophic pre σ-nowhere dense set, fuzzy neutrosophic pre σ-first category set and fuzzy neutrosophic pre σ-second category set in fuzzy neutrosophic topological spaces are introduced and studied. By means of fuzzy neutrosophic pre σ-nowhere dense sets, the concept of fuzzy neutrosophic pre....

KeywordsFuzzy neutrosophic pre-open(closed) set, fuzzy neutrosophic pre Fσ-set, fuzzy neutrosophic pre Gδ-set, fuzzy neutrosophic pre dense, fuzzy neutrosophic pre residual, fuzzy neutrosophic nowhere dense set, fuzzy neutrosophic pre σ-first and second category sets, fuzzy neutrosophic pre σ-Baire space

[1] Arockiarani I.; J. Martina Jency. More On Fuzzy Neutrosophic Sets And Fuzzy Neutrosophic Topological Spaces., International Journal Of Innovative Research & Studies, Volume 3(5), 2014, 643-652.
[2] Arockiarani I.; I. R. Sumathi.; J.Martina Jency. Fuzzy Neutrosophic Soft Topological Spaces. International Journal Of Mathematical Archive, Volume 4(10), 2013., 225-238.
[3] Chang C.L, Fuzzy Topological Spaces, Journal Of Mathematical Analysis And Applications. Volume 24(1), 1968., 182-190.
[4] Fatimah M. Mohammed1 X And Anas A. Hijab 1 And Shaymaa F. Matar1. Fuzzy Neutrosophic Weakly-Generalized Closed Sets In Fuzzy Neutrosophic Topological Spaces, J.University Of Anbar For Pure Science : Vol.12(2), 2018, 63-73.
[5] Poongothai,E.; Kalaivani, E. Neutrosophic Σ-Baire Spaces On Fuzzy Setting. Journal Of Xedian University, 2024, Volume 18(1,) Pp. 253-264..

Abstract : ....

......

Abstract : We investigate the composition operators on the space of Dirichlet series and to show the relation of Dirichlet series on Hilbert spaces, and to discuss the integral and boundry of Dirichlet series.

Keywords: Dirichlet series, Hilbert space, Drury Arveson space, polydisk, interpolation theorem, interpolating sequence, full interpolating problem, Carleson embedding, pure isometry

[1] H. Hedenmalm, P. Lindqvist And K. Seip, A Hilbert Space Of Dirichlet Series And Systems Of Dilated Functions In L2(0,1), Duke Math. J. 86, 1-37 (1997).
[2] J. E. Littlewood, On Inequalities In The Theory Of Functions Proc. London Math. Soc. 23 (1925), 481–519.
[3] H. L. Montgomery, Ten Lectures On The Interface Between Analytic Number Theory And Harmonic Analysis, Volume 84 Of Cbms Regional Conference Series In Mathematics, Ams, 1994.
[4] B. V. Gnedenko, The Theory Of Probability, 4th Ed., Chelsea, New York, 1967.
[5] E.C. Titchmarsh. The Theory Of Functions. Oxford University Press, London, 1932.

Abstract : Frieze Groups are plane symmetry groups whose translations are isomorphic to 𝑍. The word ‘frieze’ is derived from a Greek word which means decorative pattern repeated all the way around the building usually seen in Greek architecture. Similarly we have another set of groups which are an extension of frieze groups called crystallographic groups in which the translations are isomorphic to 𝑍⊕𝑍. Commonly referred as wallpaper groups, they have much more flexibility as compared to frieze groups. In this paper we shall aim to study about crystallographic groups and their symmetries.

Keywords – Frieze, Crystallographic, Rotational Symmetry, Reflections

[1]. J.A. Gallian, Abstract Algebra, 2023: P. 446-467
[2]. D. Schattschneider, Vision Of Symmetry, New York: Harry Abrams, 2004
[3]. D.S. Dummit And R.M. Foote, Abstract Algebra. John Wiley & Sons, Inc., 2023
[4]. Raymond F. Tennent, Subgroup Lattices For Crystallographic Groups. January 2005

Abstract : In the evolving digital marketing landscape, effective allocation of advertising budgets across platforms such as Google Ads and Facebook Ads has become a pivotal factor in maximizing return on investment (ROI). This study proposes a mathematical optimization model using linear and non-linear programming techniques to allocate digital ad budgets based on data-driven metrics such as impressions, click-through rates (CTR), and conversion rates. Real-world campaign data was collected and analysed, and the model was applied to determine optimal budget allocation strategies. The results demonstrate the efficacy of the model in enhancing ROI and provide practical insights for digital marketers aiming to improve performance outcomes through mathematical optimization..

Keywords: Ad Budget Allocation, ROI, Digital Marketing, Linear Programming, Non-Linear Optimization, Google Ads, Facebook Ads, CTR, Conversion Rate

[1] Berman, R., & Katona, Z. (2013). The Role Of Search Advertising In The Marketing Mix. Management Science, 59(10), 2399–2414.
[2] Chan, T. Y., & Perry, M. (2017). Budgeting For Clicks: Ad Spend Allocation Strategies For ROI Optimization. Journal Of Advertising Research, 57(3), 278–289.
[3] Chatterjee, P., & Hoffman, D. (2020). The Impact Of Optimization Algorithms In Digital Marketing: A Data-Driven Perspective. Journal Of Interactive Marketing, 49, 37–50.
[4] Dholakia, R. R. (2020). Data-Driven Marketing Strategies And Their Impact On ROI: Evidence From Digital Advertising. Journal Of Digital Marketing, 12(4), 201–213.
[5] Edelman, B., & Schwarz, M. (2010). Optimal Auction Design In Sponsored Search Markets. American Economic Journal: Microeconomics, 2(1), 1–27.

Abstract : The present paper deals with the determination of transient thermal stresses in a thick annular disc with internal heat generation. A thick annular disc is considered having zero initial temperature and arbitrary heat supply is applied on the upper and lower surface where as the fixed circular edge are at zero temperature. Here we compute the effects of internal heat generation of a thick annular disc in terms of stresses along radial direction. The governing heat conduction equation has been solved by the method of integral transform technique. The results are obtained in a series form in terms of Bessel’s functions. The results for temperature change and stresses have been computed numerically and illustrated graphically.

Keywords Thick annular disc, quasi static transient thermal stresses and internal heat generation

[1] Bhongade C. M. And Durge M. H., Some Study Of Thermoelastic Steady State Behavior Of Thick Annular Disc With Internal Heat Generation, IOSR Journal Of Mathematics, Vol.7, Issue 6, Pp 47-52, 2013.
[2] Kulkarni V. S. And Deshmukh K.C., Quasi- Static Transient Thermal Stresses In Thick Annular Disc, Sadhana, Vol. 32, Part 5, Pp 561-575, 2007.
[3] Naotake Noda, Richard B Hetnarski And Yoshinobu Tanigawa, Thermal Stresses, 2nd Edn. Pp 259-261, Taylor And Francis, New York, 2003.
[4] Nowacki W. The State Of Stresses In A Thick Circular Disk Due To Temperature Field, Bull. Acad. Polon. Sci., Ser. Sci. Techn., Vol.5, P. 227, 1957.
[5] Ozisik M. N., Boundary Value Problem Of Heat Conduction, International Text Book Company, Scranton, Pennsylvania, Pp 135-148, 1968..