Mild Balanced Neutrosophic Graphs With Application

University of New Mexico Mild Balanced Neutrosophic Graphs With Application Kishore Kumar P.K.1, S. Sangeetha2 1Professor, Department of Mathematics, Jerusalem College of Engineering, Chennai, India; [email protected] 2Assistant Professor, Department of Mathematics, Jerusalem College of Engineering, Chennai, India; [email protected] 1Correspondence: [email protected] Abstract.The framework of mild balanced neutrosophic graphs extends traditional uncertain graph models by incorporating nuanced membership functions that capture truth, indeterminacy, and falsity simultaneously. This multifaceted approach allows for more accurate representation and analysis of uncertain relationships in complex systems. The theoretical foundations laid out through rigorous theorems and propositions provide a robust base for modeling real-world scenarios where ambiguity and partial information are inherent. Moreover, the application to medicinal interactions in diabetic patients demonstrates the potential of these graphs in decision-making processes for optimizing drug combinations, thereby enhancing patient safety and therapeutic outcomes. This study bridges mathematical theory with practical healthcare challenges, highlighting the versatility and applicability of neutrosophic graph models in interdisciplinary research Keywords: Mild balanced neutrosophic graphs, Balanced neutrosophic graphs, Mild Balanced Neutrosophic subgraphs. —————————————————————————————————————————- 1. Introduction The concept of fuzziness in mathematical modeling was introduced by Lofti. A. Zadeh through his groundbreaking works on fuzzy sets [20] and fuzzy algorithms [21]. Shortly thereafter, fuzzy graphs became an active area of research, especially following Rosenfeld’s work on fuzzy relationships in cognitive modeling [16]. Recognizing that fuzziness alone may not sufficiently capture partial ignorance or conflicting information, Atanassov introduced intuitionistic fuzzy sets [4], further formalized in his text book [5], where both membership and non-membership degrees are considered simultaneously. Kishore Kumar P.K., S. Sangeetha Neutrosophic Sets and Systems, Vol. 97, 2026 Intuitionistic fuzzy graph theory was further analysed by Parvathi and Karunambigai [14], who proposed structured approaches for uncertainty networks. Later extensions such as balanced and regular intuitionistic fuzzy graphs [9,11], as well as clustering algorithms for knowledge discovery [10], enriched the field with tools for structure-based analysis under uncertainty. In parallel, neutrosophic logic which is proposed by Florentin Smarandache inspired a more generalized approach to modeling ambiguous information by incorporating three components: truth, indeterminacy, and falsity [18] membership functions. Kandasamy et al. [8] formalized neutrosophic graph theory, enabling higher expressiveness for complex decision and reasoning systems. Subsequent studies explored properties of balanced neutrosophic graphs [17]. Broumi et al. [6] investigated on the concepts of single-valued neutrosophic graphs and planar neutrosophic graphs [12]. Moreover, interval-based and bipolar extensions were developed to represent uncertain data arising from expert assessments or conflicting evidence. Bipolar fuzzy graphs, as detailed by Akram et al. [1–3], introduced a framework for modeling systems with both positive and negative tendencies. At the same time, interval-valued neutrosophic models [15,19] have been effectively applied in domains like shortest path computation, where uncertainty ranges matter. Other advancements include the incorporation of decision support indices such as Wdensity [22] and the RSM index [13] to predict links in social and knowledge networks. Classical contributions such as the Erd˝os–R´enyi model of random graphs [7] still underpin much of this work, providing a probabilistic basis for analysing network properties. Collectively, these developments mark a significant shift in graph modelling techniques from deterministic to imprecise, from binary to multi-valued logic which paves the way for robust applications in data sciences, artificial intelligence, social networks, and decision support systems. This paper builds upon this rich foundation by exploring new structural and algorithmic properties in generalized fuzzy and neutrosophic graphs, contributing to the theoretical development and practical utility of graph-based uncertainty modelling. 2. Preliminaries Definition 2.1. A single valued neutrosophic graph (SVN-graph) with underlying set V is defined to be a pair G: (A, B) where 1. The functions TA:V→[0,1], IA:V→[0,1] and FA:V→[0,1] denote the degree of truth membership, degree of indeterminate membership and degree of false membership of the element vi∈V, respectively and 0 ≤TA(vi) + IA(vi) + FA(vi)≤3 for all vi∈V. 2. The functions TB:E⊆V×V→[0,1],v,FB:E⊆V×V→[0,1] are defined by TB(vi, vj)≤TA(vi)∧TA(vi), IB(vi, vj)≤IA(vi)∨IA(vi) and FB(vi, vj)∨FA(vi)∨FA(vi) for allF(vj) which denotes the degree of true, indeterminate and false membership functions of Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 723 the edge (vi, vj)∈Erespectively., where 0 ≤TB(vi, vj) + IB(vi, vj) + FB(vi, vj)≤3 forall (vi, vj)∈E(i= 1,2, ..., n). We say A, the single valued neutrosophic vertex set of V,Bthe single valued neutrosophic edge set of E. Definition 2.2. A connected subgraph H of a neutrosophic graph G: (V, E) is called intense subgraph if (i)V(H)⊆V(G) and E(H)⊆E(G) (ii) Dt(H)≤Dt(G),Di(H)≤Di(G) and Df(H)≤Df(G) Definition 2.3. A partial SVN-subgraph of SVN-graph G= (A, B) is a SVN-graph H= (V′, E′) such that V′⊆V, where T′ A(vi)≤TA(vi), I′ A(vi)≥IA(vi), F ′ A(vi)≥FA(vi) for all vi∈Vand E′⊆E, where T′ B(vi, vj)≤TB(vi, vj), I′ B(vi, vj)≥IB(vi, vj) and F′ B(vi, vj)≥ FB(vi, vj)for all (vi, vj)∈E. Definition 2.4. Let G: (A, B) be an SVNG. Then, Gis said to be strong SVNG if TB(vjvk) = TA(vj)∧TA(vk), IB(vjvk) = IA(vj)∨IA(vk) and FB(vjvk) = FA(vj)∨FA(vk). Definition 2.5. Let G: (A, B) be an SVNG. Then, Gis said to be complete SVNG if TB(vjvk) = TA(vj)∧TA(vk), IB(vjvk) = IA(vj)∨IA(vk) and FB(vjvk) = FA(vj)∨FA(vk). Definition 2.6. A connected subgraph H of a neutrosophic graph G: (V, E) is called Feeble subgraph if (i)V(H)⊆V(G) and E(H)⊆E(G) and (ii) Dt(H)> Dt(G), Di(H)> Di(G) and Df(H)> Df(G) Definition 2.7. A neutrosophic graph G: (V, E) is called a mild balanced neutrosophic graph if all connected subgraphs of Gare intense subgraphs. Definition 2.8. Two intense neutrosophic connected subgraphs H1and H2of a neutrosophic graph G: (V, E) are called equally balanced subgraphs if (i) Dt(H)≤Dt(G), Di(H)≤Di(G)and Df(H)≤Df(G) (ii) Dt(H1)≤Dt(G),Di(H1)≤Di(G) and Df(H2)≤Df(G) (iii) Dt(H1) = Dt(H2),Di(H1) = Di(H2) and Df(H1)≤Df(H2) Definition 2.9. If Dt(Hj) = Dt(G),Di(Hj) = Di(G) and Df(Hj) = Df(G) for all possible connected subgraphs Hjof G, then the graph G: (V, E) is called a strictly balanced neutrosophic graph. Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 724 3. Mild balanced neutrosophic graphs Theorem 3.1. For a strong neutrosophic graph, D(G) = (2,2,2) and it is strictly balanced. Proof. Since all the edges of the neutrosophic graph G: (V, E) are strong tB(vjvk) = tA(vj)∧ tA(vk), iB(vjvk) = iA(vj)∨iA(vk) and fB(vjvk) = fA(vj)∨fA(vk) By definition, Dµ(G) = 2PtB(vjvk) PtA(vj)∧tA(vk) =2PiA(vj)∧iA(vk) PiA(vj)∧iA(vk) =2PfA(vj)∧fA(vk) PfA(vj)∧fA(vk) = 2 Hence D(G) = (Dt(G), Di(G), Df(G))b= (2,2,2). Also all the connected subgraphs of G: (V, E) has strong edges and hence D(H) = (2,2,2) for all subgraphs Hof G. Hence G: (V, E) is strictly balanced. Corollary 3.2. A neutrosophic graph with few strong edges can never be mild balanced. Proof. If a neutrosophic graph G has few strong edges(not all the edges), then for any connected neutrosophic subgraph Hwill have only strong edges, Dt(H) = 2, Di(H) = 2 and Df(H) = 2.Hence D(H) = (2,2,2) > D(G). Hence it may not be a mild balanced neutrosophic graph. Remark 3.3. It can be noted that subgraphs with strong edges are always feeble subgraphs of an neutrosophic graph unless it should be strong. Proposition 3.4. Union of two equally balanced connected neutrosophic subgraphs, with one or more vertices in common is also equally balanced. Proof. Let H1and H2be two equally balanced connected neutrosophic subgraphs with atleast one common vertex of a neutrosophic graph G: (V, E). Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 725 By definition, D(H1) = D(H2)≤D(G) Dt(H1) = 2P∀vjvk∈V(H1)tB(vjvk) P∀vjvk∈E(H1)(tA(vj)∧tB(vk)) =2a b Dt(H2) = 2P∀vjvk∈V(H2)tB(vjvk) P∀vjvk∈E(H2)(tA(vj)∧tB(vk)) =2c d Since Dt(H1) = Dt(H2) = 2a b Dt(H1∪H2) = 2hP∀vjvk∈V(H1)tB(vjvk) + P∀vjvk∈V(H2)tB(vjvk)i P∀vjvk∈E(H1)(tA(vj)∧tB(vk)) + P∀vjvk∈E(H2)(tA(vj)∧tB(vk)) =2(a+c) b+d=2(a+ka) b+kb =2a(k+ 1) b(k+ 1) ⇒Dt(H1∪H2) = 2a b Hence, Dt(H1∪H2) = Dt(H1) = Dt(H2). Similarly we can show that, Di(H1∪H2) = Di(H1) = Di(H2) and Df(H1∪H2) = Df(H1) = Df(H2) ⇒D(H1∪H2) = D(H1) = D(H2). Corollary 3.5. If all the possible connected subgraphs of a mild balanced neutrosophic graphs are equally balanced then the graph is strictly balanced. Proof. The proof of this corollary follows from dividing the graph into two connected subgraphs which are balanced equally. From the proposition stated above it follows that the union of two equally balanced connected neutrosophic subgraphs is equally balanced, the graph itself will be made to a strictly balanced neutrosophic graph. Proposition 3.6. Two connected neutrosophic graphs G1and G2with atleast one common vertex are intense subgraphs of the neutrosophic graph G1+G2. Proof. Let G1: (V1, E1) and G2: (V2, E2) be two connected neutrosophic graphs with atleast one common vertex. Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 726 Let Dt(H1) = 2P∀vjvk∈V(H1)tB(vjvk) P∀vjvk∈E(H1)(tA(vj)∧tB(vk)) =2a b Dt(H2) = 2P∀vjvk∈V(H2)tB(vjvk) P∀vjvk∈E(H2)(tA(vj)∧tB(vk)) =2c d Dt(G1+G2) = 2hP∀vjvk∈V1,V2tB(vjvk) + P∀vjvk∈V∗tB(vjvk)i P∀vjvk∈E1,E2(tA(vj)∧tB(vk)) + P∀vjvk∈E∗(tA(vj)∧tB(vk)) where V∗and E∗are the set of vertices and strong edges between every pair of non-common vertices G1and G2. Therefore, tB(vjvk) = tA(Vi)∧tA(vj) for all vivj∈E∗. Since we are adding a strong edge betwen all the pair of non-common vertices of G1and G2. XtB(vjvk) = XtA(vj)∧tA(vk)∀vjvk∈E∗ ∴Dt(G1+G2) = 2(a+c+x) b+d+x>2a b>2c d Dt(G1+G2)> Dt(G1) and D(G2)< D(G1+G2). Similarly, it can be shown that Di(G1+G2)> Di(G1) and D(G2)< D(G1+G2) and Df(G1+G2)> Df(G1) and D(G2)< D(G1+G2). Hence G1and G2are intense subgraphs of G1+G2. In particular, D(G1) = D(G1+G2) = D(G2) if all the graphs are strong neutrosophic graphs. Proposition 3.7. Two connected neutrosophic graphs G1and G2with atleast one common vertex are not intense subgraphs of their union.(without proof) 4. Application of Mild balanced Neutrosophic graphs Here, we apply the ideas of mild balanced neutrosophic graphs in prescribing drugs for suitable patients accordingly. Let us consider the scenario where there are 4 drugs which could be prescribed for a diabetic patient. This applications determines the efficiency of improving the mixture of drugs given to a patient with multiple illness. A graphical representation of truth membership, indeterminate membership and false membership is calculated based on medical treatments and the medicines prescribed by the physician. A pictorial representation formulated by these are then embedded into a neutrosophic graph. The following drugs have been used by an diabetic patient with various illness such as thyroid, heart disease and blood pressure. We identify the combination of drugs which would be suitable for an adult to take it as lifetime even. Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 727 We take metformin, empagliflozin, lisinopril and levothyroxine which are drugs used for diabetes, cardiac disease, blood pressure and thyroid respectively. In this application we are finding a solution for the problem where these drugs are given with limited dose to a person depending on the combination which wont affect the patients health. The following diagram shows the interaction diagram of those drugs used. Figure 1. Interaction Diagram of 4 drugs for a Diabetic Patient Figure 2. Balanced Neutrosophic Vague Graph of 4 Drugs Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 728 5. Conclusion This study not only establishes a theoretical foundation but also emphasizes the practical significance of the mild balanced neutrosophic graph model in addressing complex uncertainties in diverse domains. The framework facilitates better handling of indeterminacy and inconsistency in network data, which is crucial for real-world decision-making processes. Looking ahead, the extension of this concept to incorporate adaptive parameters will enable the modeling of more dynamic and heterogeneous systems. Applications spanning healthcare, communication networks, social interactions, and logistics stand to benefit from these advancements. Such integration paves the way for innovative tools that can predict and manage uncertainty with enhanced precision, ultimately improving the efficacy of solutions in technology and science. References 1. M. Akram. Bipolar fuzzy graphs. Information Sciences, 181(24):5548–5564, 2011. 2. M. Akram, M. G. Karunambigai, K. Palanivel, and S. Sivasankar. Balanced bipolar fuzzy graphs. Journal of Advanced Research in Pure Mathematics, 6(4):58–71, 2014. 3. M. Akram and N. Waseem. Novel applications of bipolar fuzzy graphs to decision making problems. J. Appl. Math. Comput., 56:73–91, 2016. 4. K. T. Atanassov. Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20:87–96, 1986. 5. K. T. Atanassov. Intuitionistic fuzzy sets: Theory and applications. Physica-Verlag, Heidelberg, 1999. 6. S. Broumi, F. Smarandache, M. Talea, and A. Bakali. Single valued neutrosophic graphs: degree, order and size. In Proceedings of the IEEE Conference on Fuzzy Systems, pages 2444–2451, 2016. 7. P. Erd˝os and A. R´enyi. On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5:17–61, 1960. 8. V. Kandasamy, K. Ilanthenral, and F. Smarandache. Neutrosophic graphs: a new dimension to graph theory. Infinite Study, 2015. 9. M. G. Karunambigai, M. Akram, S. Sivasankar, and K. Palanivel. Balanced Intuitionistic Fuzzy Graphs. Applied Mathematical Sciences, 7(51):2501–2514, 2013. 10. M. G. Karunambigai, M. Akram, S. Sivasankar, and K. Palanivel. Clustering algorithm for intuitionistic fuzzy graphs. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 25:367–383, 2017. 11. M. G. Karunambigai, S. Sivasankar, and K. Palanivel. Some properties of regular intuitionistic fuzzy graphs. International Journal of Mathematics and Computation, 26:53–61, 2015. 12. R. Mahapatra, S. Samanta, and M. Pal. Generalized neutrosophic planar graphs and its application. Journal of Applied Mathematics and Computing, 65(1):693–712, 2021. 13. R. Mahapatra, S. Samanta, M. Pal, and Q. Xin. RSM index: a new way of link prediction in social networks. Journal of Intelligent & Fuzzy Systems, 37(2):2137–2151, 2019. 14. R. Parvathi and M. G. Karunambigai. Intuitionistic fuzzy graphs. In Computational Intelligence: Theory and Applications, pages 139–150. Springer, 2006. 15. S. Prabha, S. Krishna, S. Broumi, and F. Smarandache. Interval valued neutrosophic shortest path problem by A* algorithm. In Proceedings of the IEEE Conference on Fuzzy Systems, 2020. Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 729 16. A. Rosenfeld. Fuzzy graphs: fuzzy sets and their applications to cognitive and decision processes. Academic Press, 1975. Pages: 95. 17. S. Sivasankar and Said Broumi. Balanced Neutrosophic Graphs. Neutrosophic Sets and Systems, 50:309–319, 2022. 18. F. Smarandache. Neutrosophic set, a generalization of the intuitionistic fuzzy set. International Journal of Pure and Applied Mathematics, 24:287–297, 2005. 19. H. Wang, F. Smarandache, Y. Q. Zhang, and R. Sunderraman. Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Phoenix, AZ, 2005. 20. L. A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965. 21. L. A. Zadeh. Fuzzy algorithms. Information and Control, 12:94–102, 1968. 22. S. Zhang, H. Sun, and X. W. Li. W-density and W-balanced property of weighted graphs. Applied Mathematics Journal of Chinese University Series B, 7(3):355–364, 2007. Kishore Kumar P.K., S. Sangeetha, Mild Balanced Neutrosophic Graphs With Applications Neutrosophic Sets and Systems, Vol. 97, 2026 730 Received: June 1, 2025. Accepted: Nov 18, 2025