we deduce from Remark 10 that G is not a {p, q}-group. Thus N is a nontrivial normal abelian Hall subgroup of G. Now N G is a { p, q } -group, so its **bipartite** **divisor** **graph** is not a cycle of length four. As cd( N G ) ⊆ cd(G), there exists no element of cd( N G ) which is a prime power. So for each nonlinear χ ∈ Irr( G N ), χ(1) = p α q β , for some positive integers α and β. This implies that G N is the direct product of its Sylow subgroups which are nonabelian. But this contradicts the form of cd( G N ). Thus

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Abstract. The concept of the **bipartite** **divisor** **graph** for integer subsets has been considered in [M. A. Iranmanesh and C. E. Praeger, **Bipartite** **divisor** graphs for integer subsets, Graphs Combin., 26 (2010) 95–105.]. In this paper, we will consider this **graph** for the **set** of **character** **degrees** of a finite group G and obtain some properties of this **graph**. We show that if G is a solvable group, then the number of connected components of this **graph** is at most 2 and if G is a non-solvable group, then it has at most 3 connected components. We also show that the diameter of a connected **bipartite** **divisor** **graph** is bounded by 7 and obtain some properties of groups whose graphs are complete **bipartite** graphs.

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Throughout this paper, let G be a finite group and let all characters be complex characters. Also, let l(G) be the largest **irreducible** **character** degree of G, s(G) be the second largest **irreducible** **character** degree of G and t(G) be the third largest **irreducible** **character** degree of G. The **set** of all **irreducible** characters of G is shown by Irr(G) and the **set** of all **irreducible** **character** **degrees** of G is shown by cd(G). In [4], B. Huppert conjectured that if G is a finite group and S is a finite non-abelian simple group such that cd(G) = cd(S), then G ∼ = S × A, where A is an abelian group. In [7], [11] and [12], it is shown that L 2 (p), simple K 3 -groups and Mathieu simple groups are determined uniquely by their

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The first step in generating the concept-based fuzzy clusters is to obtain a **set** of concepts associated with the users’ queries. The source of the conceptual information is a concept knowledge base that was originally devised for query. This concept knowledge base contains relationships between concepts and the terms have been used to describe them. The ACM Computing Classification System was used as the source of the conceptual knowledge for the prototype tool, resulting in a concept knowledge base specifically for the computer science domain. The process for obtaining the concepts that are related to the users queries is similar to the process for generating the query space as described. The query terms are first processed using Porter’s stemming algorithm, which removes the prefixes and suffixes from terms to generate the root words, called stems. These stems are matched to the stems in the concept knowledge base, and the nearest concepts are selected. For each of these concepts, the **set** of stems that are nearest to the concept are selected from the knowledge base. Each of these sets will contain one or more of the original query term stems, plus additional stems that are not present in the query. Therefore, as a result of this query space generation, a **set** of concept vectors C = { } are generated. If the total number of unique stems that were selected from the concept knowledge base is p, then the dimension of all vectors (i = 1 . . .m) is p. Further, the magnitude of the vector (i = 1 . . .m) on dimension j (j = 1 . . . p) is given by the concept knowledge base weight between concept i and term j.

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By a **graph**, we mean a finite undirected **graph** without loops or multiple edges. For standard terminology and notations related to **graph** theory we refer to Harary [3] . A labeling of **graph** is a map that carries the **graph** elements to the **set** of numbers, usually to the **set** of non-negative or positive integers. If the domain is the **set** of edges, then we speak about edge labeling. If the labels are assigned to both vertices and edges, then the labeling is called total labeling. Cordial labeling is extended to **divisor** cordial labeling, prime cordial labeling, total cordial labeling, Fibonacci cordial labeling etc.

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v ∈ V (G) be a cordial vertex of G. Then G − v is also cocomplete **bipartite** **graph**. Since δ(G) > 2, let u, w ∈ V (G), such that u, w ∈ N(v), i.e. (u, v, w) is a path in G. Since G − v is also cocomplete **bipartite** **graph**, there exists w 0 ∈ V (G − v), such that (u, w 0 , w) is a path in G − v and hence in G. Hence there are at least two paths of length two between any two neighbors of v in G.

cases (1) and (3), | G/N : M/N | is a power of 2. So suppose p = 3 and n is an odd prime. We claim that G/N ≃ P GL(2, q). Suppose it is not true. Since | G/N : M/N | is a **divisor** of 2n and n is an odd prime, we have either | G/N : M/N | = n or | G/N : M/N | = 2n. In each case we conclude that n ∈ ρ(G/N ). On the other hand, neither 3 n − 1 nor 3 n + 1 is divisible by 3, as 3 n ≡ 3(mod n). If n ̸ = 3, then ρ(G/N ) = { 2, 3, r, t, n } , a contradiction. If n = 3, then M/N ≃ P SL(2, 27). Since G/N is not isomorphic with P GL(2, 27), [6, Theorem A] verifies that 3(q − 1) and 3(q + 1) are elements of cd(G/N ). Hence 2, 3 and r generate a triangle in ∆(G/N ), a contradiction. Thus G/N ≃ P GL(2, q) which implies that |G/N : M/N | is a power of 2.

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G(R) be the **graph** of R. Defined by G(R) = (V(R), E(R)) where V(R) be the vertex **set** of G(R) and E(R) be the edge **set** of G(R), where the **set** of all the elements of ring R are consider as the vertices of **graph** G(R). For any two elements x, y∈R be considered as vertices of G(R), if x and y are adjacent in G(R). the edge **set** E(R) ={x, y∈R /x and y are adjacent iff x.y = 0, x ≠y}

Varatharajan et al. [11], introduced the concept of **divisor** cordial and proved the graphs such as path, cycle, wheel, star and some complete **bipartite** graphs are **divisor** cordial graphs and in [12], they proved some special classes of graphs such as full binary tree, dragon, corona, , and , are **divisor** cordial.

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By a **graph**, we mean a finite undirected **graph** without loops or multiple edges. For standard terminology and notations related to **graph** theory we refer to Harary [3] . A labeling of **graph** is a map that carries the **graph** elements to the **set** of numbers, usually to the **set** of non-negative or positive integers. If the domain is the **set** of edges, then we speak about edge labeling. If the labels are assigned to both vertices and edges, then the labeling is called total labeling. Cordial labeling is extended to **divisor** cordial labeling, prime cordial labeling, total cordial labeling, Fibonacci cordial labeling etc.

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Surprisingly, similar to the zero-**divisor** graphs of commutative semigroups [17, Theorem 1.3], the **graph** RG(M ), for any R-module M , is connected and the best upper-bound for diam RG(M ) is 3 if the **graph** RG(M ) is non-empty (see Corollary 3.12). Here we need to recall that the distance between two vertices in a simple **graph** is the number of edges in a shortest path connecting them. The greatest distance between any two vertices in a **graph** G is the diameter of G, denoted by diam(G) [18, p. 8].

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the number of edges incident to v. A **graph** G is complete if there is an edge between every pair of the vertices. A subset X of the vertices of a **graph** G is called independent if there is no edge with two endpoints in X. A **graph** G is called **bipartite** if its vertex **set** can be partitioned into two subsets X and Y such that every edge of G has one endpoint in X and other endpoint in Y . A **graph** G is said to be star if G contains one vertex in which all other vertices are joined to this vertex and G has no other edges. A path of length n is an ordered list of distinct vertices v 0 , v 1 , ..., v n such that v i is adjacent to v i+1 for i = 1, 2, ..., n − 1. We use

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Network protocols are a **set** of conventions followed in a network environment for initiating connections, manage communication resource stability, adoption of new nodes, discarding existing nodes and safely switching different connections based on the necessities. There are several types of protocols involved in a communication such as Basic network communication protocols, Network security protocols, Network routing protocols and Network management protocols. Here the routing protocols are used to establish connections by analyzing possible communication paths between source and destination nodes. The indent of a routing protocol can be communication speed, network stability, optimum power utilization or the combination of more than one objective. Commonly used protocols in IoT are Bluetooth protocol, WiFi IEEE 802.11 b/g/n, MQTT, CoAP, DDS, AMQP, LoRa and Zigbee.

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A **graph** ( , ) is of vertices and edges. The vertex **set** ( ) is non-empty **set** and the edge **set** ( ) may be empty. Labelings of graphs subject to certain condition gave raise to enormous work which listed by J. A. Gallian [1], Cube **Divisor** Cordial **Graph** were introduced by K. K. Kanani and M. I. Bosmia [2] . Let = ( ( ), ( )) be a simple **graph** and ∶ ( ) → 1, 2, … , | ( )| be a bijection. For each edge = , assign the label 1 if [ ( )] / ( ) or [ ( )] / ( ) and the label 0 otherwise. The function f is called a Cube **Divisor** Cordial Labeling if | (0) − (1)| ≤ 1. S. K. Vaidya and U. M. Prajapati introduced ⊕ admits some results on prime and K - prime labeling [5] , The **graph** ⊕ introduced S. K. Vaidya and U. M. Prajapati on some results on prime and K-prime labeling [5] , A. Solairaju and R. Raziya Begam proved the merge **graph** ∗ on edge-magic labeling of some graphs [3] , the bow **graph** , + proved R. Uma and N. Arun vigneshwari on Square sum labeling bow, bistar,

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The cycle structure of Cayley graphs and Unitary Cayley graphs were studied by Berrizbeitia and Guidici [1, 2] and Detzer and Guidici [6]. Recently Maheswari and Madhavi [8– 10] studied the enumeration methods for finding the number of triangles and Hamilton cycles in arithmetic graphs associated with the quadratic residues modulo a prime p and the Euler totient function ϕ(n) , n ≥ 1 an integer. In [4] Chalapathi et al. gave a method of enumeration of triangles in the arithmetic Cayley **graph**, namely the **divisor** Cayley **graph** associated with the **divisor** function d(n), n ≥ 1 an integer. The main aim of this paper is to give an enumeration process for counting the number of disjoint Hamilton cycles in the **divisor** Cayley **graph**. In this study we have followed Bondy and Murty [3] for **graph** theory and Apostol [13] for number theory terminology.

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Since a specialist group is usually heterogeneous, opinions cannot be considered with the same degree of importance. The determination of the degree of importance of the specialist is done by means of a data collection instrument. This instrument used for data collection is a questionnaire that was used by BELCHIOR (1997) and MORÉ (2004) to identify the profile of the specialist. Each questionnaire contains information from a GIE single specialist. The respective **degrees** of importance is defined as a subset µi (k) Є [0,1]. The degree of importance of each specialist, GIEi, which is their relative degree of importance compared to other specialists, is defined by:

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sentially a sequence of weights for edges between u, v ∈ P and each node in L u,v . Similarity of two nodes is measured using the cosine similarity of their corresponding sequences, k~akk~bk ~a· ~b , which we compute using a function sim(·, ·). Thus, doc- ument groups are considered more similar when they have similar sets of topics in similar propor- tions. As we will show later, this simple solution, based on item-based collaborative filtering (Sar- war et al., 2001), is surprisingly effective at infer- ring similarity among document groups in G. Node Clusters. Identifying clusters of related nodes in the **bipartite** **graph** G can show how doc- ument groups form larger classes. However, we find that G is typically fairly dense. For these reasons, partitioning of the one-mode projection of G and other standard **bipartite** **graph** cluster- ing techniques (e.g., Dhillion (2001) and Sun et al. (2009)) are rendered less effective. We instead employ a different tack and exploit the node sim- ilarities computed earlier. We transform G into a new weighted **graph** G P = (P, E P , w sim ) where E P = {(u, v) | u, v ∈ P, sim(u, v) > ξ} , ξ

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Definition 2.1. Let G = ( V , E ) be a **graph**. A mapping f : V → {0,1} is called the binary vertex labeling of G and f (v ) is called the label of the vertex v ∈ V of G under f. The induced edge labeling f * : E → {0,1} is given by f * ( e ) =| f ( u ) − f ( v ) |, for all e = uv ∈ E .

[ where G [ ] is the vertex induced sub **graph** of G. The neighbourhood number of G is defined as the minimum cardinality of a neighbourhood **set** S of G[1]. The degree of a vertex V in an interval **graph** G is the number of edges of G incident with V and it is denoted by degree of V that is deg(v). The maximum or the minimum degree among the vertices of G is denoted by ∆( ) or ( ). In this connection we will consider the maximum degree of vertices v from G corresponding to I

In this paper, we study the undirected **graph** Γ 𝐼𝐼 (𝑀𝑀) of Gamma near rings for any completely reflexive ideal I of M. Throughout this paper M stands for a non zero Gamma near -ring with zero element and I is a completely reflexive ideal of M. For distinct vertices x and y of a **Graph** G, let d(x, y) be the length of the shortest path from x to y. The diameter of a connected **graph** is the supremum of the distances between vertices. For any **graph** G, the girth of G is the length of a shortest cycle in G and is denoted by gr(G). If G has no cycle, we define the girth of G to be infinite. A clique of a **graph** is a maximal complete subgraph and the number of **graph** vertices in the largest clique or **graph** G, denoted by ω(G) is called the clique number of G. A **graph** G is **bipartite** with vertex classes 𝑉𝑉 1 , 𝑉𝑉 2 if the **set** of all vertices of G is 𝑉𝑉 1 ∪ 𝑉𝑉 2 , 𝑉𝑉 1 ∩ 𝑉𝑉 2 = ∅ , and edge of G joins a vertex from 𝑉𝑉 1 to a vertex of 𝑉𝑉 2 .

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