Top PDF Bipartite divisor graph for the set of irreducible character degrees

Bipartite divisor graph for the set of irreducible character degrees

Bipartite divisor graph for the set of irreducible character degrees

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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On bipartite divisor graph for character degrees

On bipartite divisor graph for character degrees

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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Characterization of some simple $K_4$-groups by some irreducible complex character degrees

Characterization of some simple $K_4$-groups by some irreducible complex character degrees

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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Personalized Concept and Fuzzy Based Clustering of Search Engine Queries

Personalized Concept and Fuzzy Based Clustering of Search Engine Queries

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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Sum Divisor Cordial Labeling of Herschel Graph

Sum Divisor Cordial Labeling of Herschel Graph

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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Vol 2013

Vol 2013

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.

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On nonsolvable groups whose prime degree graphs have four vertices and one triangle

On nonsolvable groups whose prime degree graphs have four vertices and one triangle

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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Vol 6, No 11 (2015)

Vol 6, No 11 (2015)

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}

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Some New Families of Divisor Cordial Graph

Some New Families of Divisor Cordial Graph

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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Sum Divisor Cordial Labeling of Theta Graph

Sum Divisor Cordial Labeling of Theta Graph

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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A generalization of zero-divisor graphs

A generalization of zero-divisor graphs

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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$C_4$-free zero-divisor graphs

$C_4$-free zero-divisor graphs

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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Strengthening IoT WSN Architecture for Environmental Monitoring

Strengthening IoT WSN Architecture for Environmental Monitoring

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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Cube Divisor Cordial Labeling For Some Graphs

Cube Divisor Cordial Labeling For Some Graphs

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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Enumeration of disjoint Hamilton cycles in a divisor Cayley graph

Enumeration of disjoint Hamilton cycles in a divisor Cayley graph

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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A Fuzzy Logic-Possibilistic Methodology to Analyze the Main Corrosion Damages Mechanisms in Pipes and Equipment Installed in an Oil and Gas Platform

A Fuzzy Logic-Possibilistic Methodology to Analyze the Main Corrosion Damages Mechanisms in Pipes and Equipment Installed in an Oil and Gas Platform

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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A Framework for Comparing Groups of Documents

A Framework for Comparing Groups of Documents

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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Further Results on Vertex Odd Divisor Cordial Labeling of Some Graphs

Further Results on Vertex Odd Divisor Cordial Labeling of Some Graphs

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 .

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The Comparison of Neighbourhood Set and Degrees of an Interval Graph G Using an Algorithm

The Comparison of Neighbourhood Set and Degrees of an Interval Graph G Using an Algorithm

[ 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

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Vol 4, No 12 (2013)

Vol 4, No 12 (2013)

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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