# Paley Graphs and Their Generalization: Master Thesis, 2009

## Ahmed Noubi Elsawy

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To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We study some important properties of the Paley graphs. In particular, we show that the Paley graphs are connected, self-complementary, and symmetric. Also we show that the Paley graph of order q is (q-1)/2 regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors. In other words, the Paley graphs are strongly regular with parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. We give three examples of these generalizations and some of their basic properties. We also define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge iff there difference belongs to the m-th power of the multiplicative group of the field, for any odd integer m < 1, and we call them the m-Paley graphs. We show that the m-Paley graph of order q is complete iff gcd(m, q - 1)=1 and when d = gcd(m, q - 1) < 1, the m-Paley graph is (q-1)/d regular.

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I am an egyptian mathematician. I had my Bachelor from Al-Azhar University, Cairo-Egypt. I got a scholarship from the egyptian government to obtain my PhD from Germany. I wrote my Master Thesis as a part of a fast track Phd program in the University of Dusseldorf, Germany. In 2012 I got my Phd. under supervision of Prof. Oleg Bogopolski.

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## 1.Paley Graphs and Their Generalization

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Book Description Condition: New. Publisher/Verlag: LAP Lambert Academic Publishing | Master Thesis, 2009 | To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We study some important properties of the Paley graphs. In particular, we show that the Paley graphs are connected, self-complementary, and symmetric. Also we show that the Paley graph of order q is (q-1)/2 regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors. In other words, the Paley graphs are strongly regular with parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. We give three examples of these generalizations and some of their basic properties. We also define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge iff there difference belongs to the m-th power of the multiplicative group of the field, for any odd integer m 1, and we call them the m-Paley graphs. We show that the m-Paley graph of order q is complete iff gcd(m, q - 1)=1 and when d = gcd(m, q - 1) 1, the m-Paley graph is (q-1)/d regular. | Format: Paperback | Language/Sprache: english | 52 pp. Seller Inventory # K9783848442362

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Book Description LAP Lambert Academic Publishing Mrz 2012, 2012. Taschenbuch. Condition: Neu. Neuware - To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We study some important properties of the Paley graphs. In particular, we show that the Paley graphs are connected, self-complementary, and symmetric. Also we show that the Paley graph of order q is (q-1)/2 regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors. In other words, the Paley graphs are strongly regular with parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. We give three examples of these generalizations and some of their basic properties. We also define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge iff there difference belongs to the m-th power of the multiplicative group of the field, for any odd integer m 1, and we call them the m-Paley graphs. We show that the m-Paley graph of order q is complete iff gcd(m, q - 1)=1 and when d = gcd(m, q - 1) 1, the m-Paley graph is (q-1)/d regular. 52 pp. Englisch. Seller Inventory # 9783848442362

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Book Description LAP Lambert Academic Publishing Mrz 2012, 2012. Taschenbuch. Condition: Neu. Neuware - To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We study some important properties of the Paley graphs. In particular, we show that the Paley graphs are connected, self-complementary, and symmetric. Also we show that the Paley graph of order q is (q-1)/2 regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors. In other words, the Paley graphs are strongly regular with parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. We give three examples of these generalizations and some of their basic properties. We also define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge iff there difference belongs to the m-th power of the multiplicative group of the field, for any odd integer m 1, and we call them the m-Paley graphs. We show that the m-Paley graph of order q is complete iff gcd(m, q - 1)=1 and when d = gcd(m, q - 1) 1, the m-Paley graph is (q-1)/d regular. 52 pp. Englisch. Seller Inventory # 9783848442362

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## 8.Paley Graphs and Their Generalization (Paperback)

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Book Description LAP Lambert Academic Publishing, Germany, 2012. Paperback. Condition: New. Aufl.. Language: English . Brand New Book. To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We study some important properties of the Paley graphs. In particular, we show that the Paley graphs are connected, self-complementary, and symmetric. Also we show that the Paley graph of order q is (q-1)/2 regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors. In other words, the Paley graphs are strongly regular with parameters(q, q-1/2, q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. We give three examples of these generalizations and some of their basic properties. We also define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge iff there difference belongs to the m-th power of the multiplicative group of the field, for any odd integer m. Seller Inventory # KNV9783848442362

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## 9.Paley Graphs and Their Generalization : Master Thesis, 2009

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Book Description LAP Lambert Academic Publishing Mrz 2012, 2012. Taschenbuch. Condition: Neu. This item is printed on demand - Print on Demand Neuware - To construct a Paley graph, we fix a finite field and consider its elements as vertices of the Paley graph. Two vertices are connected by an edge if their difference is a square in the field. We study some important properties of the Paley graphs. In particular, we show that the Paley graphs are connected, self-complementary, and symmetric. Also we show that the Paley graph of order q is (q-1)/2 regular, and every two adjacent vertices have (q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common neighbors. In other words, the Paley graphs are strongly regular with parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many mathematicians. We give three examples of these generalizations and some of their basic properties. We also define a new generalization of the Paley graphs, in which pairs of elements of a finite field are connected by an edge iff there difference belongs to the m-th power of the multiplicative group of the field, for any odd integer m 1, and we call them the m-Paley graphs. We show that the m-Paley graph of order q is complete iff gcd(m, q - 1)=1 and when d = gcd(m, q - 1) 1, the m-Paley graph is (q-1)/d regular. 52 pp. Englisch. Seller Inventory # 9783848442362

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## 10.Paley Graphs and Their Generalization

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