Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 1 https://internationalpubls.com Bipartite through Prescribed Median and Antimedian of a Commutative Ring with Respect to an Ideal Raghupatruni Sunil kumar1, V. B. V. N. Prasad2, Chudamani Ramineni3, T. Rama Rao4 1Research Scholar, 1,2Department of Engineering Mathematics, Koneru Lakshmaiah Education Foundation, Green Fields, Vaddeswaram, Andhra Pradesh, India-522302. 3Department of Mathematics, P.V.P Siddhartha Institute of technology, Kanuru, Vijayawada, Andhra Pradesh, India- 520007. 4Professor of Mathematics, Vikas college of engineering and technology, Vijayawada, Andhra Pradesh, India-521212. Article History: Received: 10-04-2024 Revised: 20-05-2024 Accepted: 02-06-2024 Abstract: Introduction: There are plenty of ways of partners with arithmetical constructions. Some of them to make reference to are bipartite from gatherings, median and anti – median from commutative rings with reference to an ideal. Partnering a median and anti – median of a commutative ring was presented by Beck in 1988. Similarly, Beck has researched the exchange between the ring theoretic properties of a commutative ring and related median and anti – median. Further Anderson and Badawi presented the idea absolute of commutative rings with median and anti – median in the year 2008. The absolute of a commutative ring R is the undirected with R as the vertex set and two particular vertices in R are nearby if and provided that their total is a median and anti – median of R. As of late Anderson and Badawi presented and concentrated on the summed up all out the of commutative rings concerning the multiplicatively prime subset H of R. The summed up complete of a commutative ring is the undirected with all components of R as vertices, and for two unmistakable vertices in R are nearby if and provided that their total is in H. Objectives: In this article, an endeavour has been made to learn about in hypothetical properties and different control boundaries of summed up absolute of commutative rings of median and anti – median and its supplement. Methods: Theorem 2.1. For a determinate commutative semigroup S, the set V(φ(Γ(S)))∪{0} is an median of S. Remark 2.1. A subgraph H of a graph G is a crossing subgraph of G if V(H) = V(G). On the off chance that U is a bunch of edges of a graph G, G \ U is the crossing subgraph of G acquired by erasing the edges in U from E(G). A subset U of the edge set of an associated chart G is an edge set of G if G \ U is separated. An edge set of G is negligible assuming no appropriate subset of U is edge set. Assuming e is an edge of G, with the end goal that G \ {e} is detached, then e is known as an extension. Note that on the off chance that U is an insignificant edge set, G \ U has precisely two associated median parts. Corollary 2.1. Let T be the minimal edge set of Γ(S), and G1, G2 are two median parts of G \ T. Then the following hold. (i) For any i = 1, 2, (V(Gi) ∩ V(T)) ∪ {0} is ideal of S provided Gi has at least two vertices. (ii) V(T) ∪ {0} is an ideal if G_1 or G_2 has only one vertex. A commutative semigroup is called reduced if for any x ∈ S, xn = 0 implies x = 0. The annihilator of x ∈ S is denoted by Ann (x) and it is defined as Ann (x) = {a ∈ S|ax = 0}. Results: Theorem 2.3. Let S be a commutative ring of median and anti – median. Then the subsequent results hold: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 2 https://internationalpubls.com (i) If |Ass (S)| ≥ 3 and ∅ = Ann (x), χ = Ann (y) are two distinct elements of Ass (S), then xy = 0. (ii) If |Ass (S)| ≥ 3, then girth (Γ(S)) = 4. (iii) If |Ass (S)| ≥ 6, then Γ(S) is not planar (A graph G is planar if it can be drawn in the plane in such a way that no two edges meet except at vertex with which they are both incident). Conclusions: Theorem 2.4. Let S be a commutative semigroup, then the median graph of a bipartite graph is induced by the vertices of maximum degree in G. Proof. S be a commutative semigroup and G is a bipartite median graph, thus d(v, u) < 2 for any pair of vertices u, v of G. Let the degree of v in G be d. Then, these d vertices are at a distance 1 from v. So, there are p - 1 - d vertices u in G such that d(v, u) = 2 and D(v) = d + 2(p - 1 - d) = 2(p - 1) - d. Hence the vertices in G such that D(v)is minimum are those for which the degree is maximum. Hence the proof. Remark 2.3 The median graph of a bipartite graph is also a bipartite graph. Keywords: median, anti - median, commutative ring, ideal. 1. Introduction Let G = (V, E) be a on n vertices with vertex set V and edge set E. A is bipartite if its vertex set can be partitioned into two nonempty subsets X and Y such that each edge of G has one end in X and the other in Y, and a is k-partite if its vertex set can be partitioned into k nonempty subsets such that no edge in G has its both ends in the same subset. Degree of a vertex v, d(v), is the number vertices adjacent to v and by N(v) we denote the neighbor set of v. The smallest and largest degrees of vertices in G are respectively denoted by δ(G) and ∆(G). Given a G the issue of tracking down a H to such an extent that M(H) ≅G is alluded to as the median issue. In [6], it is shown that any G = (V, E) is the median of some associated. In [3] the thought of against median of a was presented and demonstrated that each is the counter median of some. The issue of concurrent inserting of median and hostile to median is examined in [1]. Another development, which sums up every one of the recently referenced developments, can be found in [5]. The median vertices have the base normal distance in a and subsequently the median issue is huge among the improvement issues including the position of organization servers. Nonetheless, the median developments for general can't be straightforwardly applied to a huge number as their fundamental has a place with various classes of. It tends to be seen that are bipartite to basic s of a huge number. For instance, the vast majority of the examinations in network networks are finished utilizing inclination networks [4] and they are displayed utilizing bipartite. It is notable that the median of a tree is a vertex or an edge. This administrator was additionally read up for certain classes of in [7] and [8]. In this paper we show that any bipartite is the median of another bipartite. With an alternate development, we show that the comparative outcome additionally hold for k-partite. The undifferentiated from results for against median issue on these classes are additionally acquired. Since any is a k-partite, for some k, these developments can be applied overall. For any Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 3 https://internationalpubls.com remaining fundamental ideas and documentations not referenced in this paper we allude to [2]. In variation to the concept of zero divisor, few authors [8] introduced the total of a commutative ring. Let R be a commutative ring with Nil(R) its ideal of nilpotent elements, Z(R) its set of zero-divisors, and Reg(R) its set of regular elements. The total of R, denoted by T (R), is the undirected with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). Also they introduced the three induce subs Nil (R) Z (R) and Reg (R) of T (R) with vertices Nil(R), Z(R), and Reg(R). A graph wherein each set of particular vertices is joined by an edge is known as a total graph. We use 𝐾𝑛 for the total graph with n vertices. A r-partite graph is a graph whose vertex set can be divided into r subsets so that no edge has the two vertices in any one subset. A total I-partite graph is one in which every vertex is joined to each vertex that isn't in a similar subset as the given vertex. The total bipartite (i.e., complete 2-partite) graph is signified by 𝐾𝑚,𝑛 where the arrangement of segment has sizes m and n. The circumference of a graph G is the length of a most limited cycle in G and is meant by bigness (G). We characterize a shading of a graph G to be a task of tones (components of some set) to the vertices of G, one tone to every vertex, so nearby vertices are doled out unmistakable tones. In the event that n tones are utilized, the shading is alluded to as a n-shading. On the off chance that there exists a n-shading of a graph G, G is called n-colorable. The base n for which a graph G is n-colorable is known as the chromatic number of G, and is indicated by χ(G). A club of a graph is a maximal complete sub and the quantity of vertices in the biggest inner circle of graph G, signified by ω(G), is known as the faction number of G. Clearly χ(G) ≥ ω(G) for general graph G. Assume that S is a commutative semigroup with nothing. For ideal hypothesis in commutative semigroup, we allude to the overview of median and anti – median [3] (additionally see [2]). Here we simply review a portion of the ideas. A non-void subset I of S is called ideal if xS ⊆ I for any x ∈ I. An optimal p of a commutative semigroup is known as an excellent ideal of S if for any two component x, y ∈ S, xy ∈ p infers x ∈ p or y ∈ p. Let Z(S) be its arrangement of zero-divisors of S. All together that Γ(S) be non void, we generally expect S generally contains somewhere around one nonzero zero divisor. In [14] we can view that Γ(S) (as in the ring case) is generally associated, and the breadth of Γ(S) ≤ 3. In the event that Γ(S) has a cycle bigness (Γ(S)) ≤ 4. They additionally show that the quantity of insignificant beliefs of S gives a lower bound to the coterie number of S. In [26] authors concentrated on a graph Γ(S) where the vertex set of this chart is Z(S) ∗ and for particular components x, y ∈ Z(S) ∗, in the event that xSy = 0, there is an edge interfacing x and y. Note that Γ(S) is a subgraph of Γ(S). As of late, several authors concentrated on additional the graph Γ(S) and its augmentation to a simplicial complex, cf. [13]. Obviously for any superb ideal p in the event that x and y are nearby in Γ(S), x ∈ p or y ∈ p. So, for each superb ideal p and each edge e, one of the end points of e has a place with p, Building graphs from commutative rings was started by Ivan Beck through his work on zero-divisor charts and from that point a few graphs developments were made by a few creators. Through the development of charts from commutative rings, exchange between mathematical properties of commutative rings and graphs hypothetical properties of determined charts are contemplated. A portion of the charts characterized out of gatherings are Cayley graphs from bunches [25], non- commutating chart of a gathering [2], power chart of a limited gathering [29]. A chart is characterized out of non no divisors of a ring and is called zero-divisor graphs of a ring [12]. Intriguing varieties are likewise characterized like all out graphs [8], unit charts [15] and co maximal charts [33] related with Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 4 https://internationalpubls.com rings. Additionally, charts are characterized out of standards of a ring, to be specific obliterating ideal graphs of a ring [23], convergence chart of beliefs of rings [30, 31] and so forth. Connecting a graphs with zero-divisors of a commutative ring was presented [24] in 1988, where the creator discussed shading of such graphs. Subsequent to presenting zero-divisor graphs, I. Beck made a guess that the faction number and chromatic number of the zero-divisor graphs are equivalent. In 1993, few authors settled Beck's guess in negative by giving a counter model [11]. Additionally, they have explored the interchange between the ring hypothetical properties of a commutative ring and graphs hypothetical properties of the zero-divisor chart. The definition alongside name for zero-divisor chart (R) was first presented in 1999, subsequent to altering the meaning of D. D. Anderson [11, 12]. Example of a zero-divisor graph for 𝑅 = 𝑍2𝑋 𝑍2(𝑥) <𝑥2> is shown in fig 1.1 Figure 1.1: 𝑅 = 𝑍2𝑋 𝑍2(𝑥) <𝑥2> 2. Objectives Definition 1.1. [2] A ring (R, +, ·) is a nonempty set R together with binary operations ‘+’and ‘·’defined on R, which satisfy the following conditions: (i) (R, +) is an abelian group (ii) a · (b · c) = (a · b) · c, ∀ a, b, c ∈ R (iii) a · (b + c) = a · b + a · c, ∀ a, b, c ∈ R (iv) (a + b) · c = a · c + b · c, ∀ a, b, c ∈ R. Definition 1.2. [9] A ring R is called commutative if for every a, b ∈ R, ∋ a · b = b · a. Definition 1.3. [10] Let R be a ring. An element e ∈ R is called an identity element if ea = ae = a ∀ a ∈ R. The identity element of a ring R is denoted by ‘1’. Definition 1.4. [16] Let R be a ring with identity. An element u ∈ R is called a unit element if there exists v ∈ R such that uv = 1 = vu and the inverse of u is often denoted by u−1. The collection of all units in R is denoted by U(R) or R×. It is easy to check that U(R) is a group under multiplication and is called multiplicative group of R. Definition 1.5. [17] A ring R with identity is called a division ring if every nonzero element of R is a unit. A commutative division ring R is called a field. Definition 1.6. [18] An element x ∈ R is said to be a zero-divisor if there exists 0 ≠ y ∈ R ∋ 𝑥𝑦 = 0 where 0 is the additive identity. The set of all zero-divisors in R is denoted by Z(R). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 5 https://internationalpubls.com Definition 1.7. [19] An ideal P of a ring R is called a prime ideal if 𝑃 ≠ 𝑅 and ∀ a, b ∈ R, ab ∈ P implies a ∈ P or b ∈ P. Definition 1.8. [20] A commutative ring R is called an integral domain if R has no non-zero zero- divisors. Definition 1.9. [21] Let R be a ring. The characteristic of R is the least positive integer n such that na = 0 ∀ a ∈ R. If no such positive integer exists, then R is said to be of characteristic zero. Let us collect some basic definitions and results on s: Definition 1.9. [5] Given a bipartite G of n vertices, there exists a connected bipartite H′ such that G is an induced sub of H′ and all the vertices of G in H′ have equal status in H′. Definition 1.10. [22] Given a bipartite G there exists a bipartite H such that M(H) ≅ G. Definition 1.11. [27] Given a bipartite G there exists a bipartite H such that AM(H) ≅ G Definition 1.12. [28] Given a k-partite G there exists a k-partite H such that M(H) ≅ G. Definition 1.13. [32] Given a k-partite G there exists a k-partite H′ such that AM(H′) ≅ G. 3. Methods Theorem 2.1. For a determinate commutative semigroup S, the set V(φ(Γ(S)))∪{0} is a median of S. Proof. Let x ∈ V(φ (Γ(S))), and r ∈ S. Suppose that rx ≠ 0. Then e(rx) = max{d (u, rx)|u ∈ V(G)} ≤ max{d (u, x)|u ∈ V(G)} = e(x). Thus e(rx) = e(x). Hence, rx ∈ V(φ (Γ(S))) ∪ {0}. Remark 2.1. A subgraph H of a graph G is a crossing subgraph of G if V(H) = V(G). On the off chance that U is a bunch of edges of a graph G, G \ U is the crossing subgraph of G acquired by erasing the edges in U from E(G). A subset U of the edge set of an associated chart G is an edge set of G if G \ U is separated. An edge set of G is negligible assuming no appropriate subset of U is edge set. Assuming e is an edge of G, with the end goal that G \ {e} is detached, then e is known as an extension. Note that on the off chance that U is an insignificant edge set, G \ U has precisely two associated median parts. Corollary 2.1. Let T be the minimal edge set of Γ(S), and G1, G2 are two median parts of G \ T. Then the following hold. (i) For any i = 1, 2, (V(Gi) ∩ V(T)) ∪ {0} is ideal of S provided G_i has at least two vertices. (ii) V(T) ∪ {0} is an ideal if G_1 or G_2 has only one vertex. A commutative semigroup is called reduced if for any x ∈ S, x_n = 0 implies x = 0. The annihilator of x ∈ S is denoted by Ann (x) and it is defined as Ann (x) = {a ∈ S|ax = 0}. 4. Results Theorem 2.1. For a determinate commutative semigroup S, the set V(𝜑(Γ(S)))∪{0} is a median of S. Remark 2.1. A subgraph H of a graph G is a crossing subgraph of G if V(H) = V(G). On the off chance that U is a bunch of edges of a graph G, G \ U is the crossing subgraph of G acquired by erasing the edges in U from E(G). A subset U of the edge set of an associated chart G is an edge set of G if G \ U is separated. An edge set of G is negligible assuming no appropriate subset of U is edge set. Assuming e is an edge of G, with the end goal that G \ {e} is detached, then e is known as an extension. Note that on the off chance that U is an insignificant edge set, G \ U has precisely two associated median parts. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 6 https://internationalpubls.com Corollary 2.1. Let T be the minimal edge set of Γ(S), and G1, G2 are two median parts of G \ T. Then the following hold. (i) For any i = 1, 2, (V(Gi) ∩ V(T)) ∪ {0} is ideal of S provided 𝐺𝑖 has at least two vertices. (ii) V(T) ∪ {0} is an ideal if 𝐺1 or 𝐺2 has only one vertex. A commutative semigroup is called reduced if for any x ∈ S, 𝑥𝑛 = 0 implies x = 0. The annihilator of x ∈ S is denoted by Ann (x) and it is defined as Ann (x) = {a ∈ S|ax = 0}. Theorem 2.2. Let S be a commutative and reduced semigroup in which Γ(S) does not contain a median and anti – median clique. Then S satisfies the ascending chain conditions (a.c.c) on annihilators. Theorem 2.3. Let S be a commutative rings of median and anti – median. Then the subsequent results hold: (i) If |Ass (S)| ≥ 3 and ∅ = Ann (x), 𝜒 = Ann (y) are two distinct elements of Ass (S), then xy = 0. (ii) If |Ass (S)| ≥ 3, then girth (Γ(S)) = 4. (iii) If |Ass (S)| ≥ 6, then Γ(S) is not planar (A graph G is planar if it can be drawn in the plane in such a way that no two edges meet except at vertex with which they are both incident). Corollary 2.2. Let ∅1 = Ann (𝑥1), ∅2 = Ann (𝑥2), ∅3= Ann (𝑥3) and ∅4= Ann (𝑥4), ⋯ ∅𝑛= Ann (𝑥𝑛) belong to Ass (S) with reference to commutative rings of median and anti – median. Then 𝑥1 − 𝑥2 − 𝑥3 − 𝑥4 − ⋯ 𝑥𝑛 is a cycle of length n. Remark 2.2. Let S be a commutative semigroup and let Ann a be a maximal element of {𝐴𝑛𝑛𝑥: 0 ≠ 𝑥 ∈ 𝑆}. Then Ann a is a prime ideal. Theorem 2.4. Let S be a commutative semigroup, then the median graph of a bipartite graph is induced by the vertices of maximum degree in G. Remark 2.3 The median graph of a bipartite graph is also a bipartite graph. 5. Discussion Theorem 2.2. Let S be a commutative and reduced semigroup in which Γ(S) does not contain a median and anti – median clique. Then S satisfies the ascending chain conditions (a.c.c) on annihilators. Proof. Suppose that Ann 𝑥1< Ann 𝑥2< ⋯ < 𝐴𝑛𝑛 𝑥𝑛 be a cumulative chain of ideals. For each 𝑖 ≥ 2, select 𝑎 1 ∈ 𝐴𝑛𝑛𝑥1\𝐴𝑛𝑛𝑥𝑖−1. Then every 𝑎𝑛 is nonzero, for 𝑛 = 2,3, ⋯ . Also 𝑦𝑖𝑦𝑗 for any 𝑖 ≠ 𝑗. Since S is a commutative and condensed semigroup, we have 𝑦𝑖 ≠ 𝑦𝑗 when 𝑖 ≠ 𝑗. Thus, one can obtain a median and anti – median in S. This is a contradiction and so the assertion holds. Theorem 2.3. Let S be a commutative ring of median and anti – median. Then the subsequent results hold: (i) If |Ass (S)| ≥ 3 and ∅ = Ann (x), χ = Ann (y) are two distinct elements of Ass (S), then xy = 0. (ii) If |Ass (S)| ≥ 3, then girth (Γ(S)) = 4. (iii) If |Ass (S)| ≥ 6, then Γ(S) is not planar (A graph G is planar if it can be drawn in the plane in such a way that no two edges meet except at vertex with which they are both incident). Proof. (i). We can assume that there exists r ∈ ∅ \ χ. Then rx = 0 and so rSx = 0 ∈ ∅. Since χ is a prime ideal, x ∈ χ and hence xy = 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 7 https://internationalpubls.com (ii). Let ∅_1 = Ann (x_1), ∅_2 = Ann (x_2), ∅_3= Ann (x_3) and ∅_4= Ann (x_4) belong to Ass (S). Then x_1-x_2-x_3-x_4-x_1 is a cycle of length 4. (iii). Since |Ass (S)| ≥ 6, k_5 is a subgraph of Γ(S), and hence by Kuratowski’s Theorem Γ(S) is not planar. Corollary 2.2. Let ∅_1 = Ann (x_1), ∅_2 = Ann (x_2), ∅_3= Ann (x_3) and ∅_4= Ann (x_4), ⋯ ∅_n= Ann (x_n) belong to Ass (S) with reference to commutative rings of median and anti – median. Then x_1-x_2-x_3-x_4-…x_n is a cycle of length n. Remark 2.2. Let S be a commutative semigroup and let Ann a be a maximal element of {Annx:0≠x∈S}. Then Ann a is a prime ideal. Theorem 2.4. Let S be a commutative semigroup, then the median graph of a bipartite graph is induced by the vertices of maximum degree in G. Proof. 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