# Some New Families of Integral Trees of Diameter Four

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 2011 ¼ 6 «Æ Æ Æ 15 ² Æ 2 June, 2011 Operations Research Transactions Vol.15 No.2 Some New Families of Integral Trees of Diameter Four WANG Ligong 1 ZHANG Zheng 1 Abstract An integral graph is a graph of which all the eigenvalues of its adjacency matrix are integers. This paper investigates integral trees of diameter 4. Many new classes of such integral trees are constructed infinitely by solving some certain Diophantine equations. These results generalize some results of Wang, Li and Zhang (see Families of integral trees with diameters 4, 6 and 8, Discrete Applied Mathematics, 2004, 136: ). Keywords Operations research, integral tree, characteristic polynomial, diophantine equation, graph spectrum Subject Classification (GB/T ) Ú Ô 4 ÙÓÕÒ ÝÜÛ 1 Þ ß 1 Ø ¹ È 4»Ä Å ±ÎÀ É Å ³ µæ ( Families of integral trees with diameters 4, 6 and 8, Discrete Applied Mathematics, 2004, 136: ) ËÄ º «Æ Ã Å ¾ ÖÐ Ñ (GB/T ) Introduction Throughout this paper we shall consider only simple graphs (i.e. undirected graphs without loops and/or multiple edges). Let G be a simple graph with vertex set V (G) = {v 1,v 2,,v n } and edge set E(G). The adjacency matrix A(G) = [a ij ] of G is an n n symmetric matrix of 0 s and 1 s with a ij = 1 if and only if v i and v j are joined by an edge. The characteristic polynomial of G is the polynomial P(G,x) = det(xi n A(G)), where I n always denotes the n n identity matrix. The 2009 ½ 1 6 Supported by the NNSF of China (No ), the Natural Science Foundation of Shaanxi Province (No. SJ08A01) and SRF for ROCS, SEM. 1. Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi an , China; Á Ê Ç ÇÏÌÍÇÂ Á

2 2 WANG Ligong, ZHANG Zheng 15 ² spectrum of A(G) is also called the spectrum of G. The research on integral graphs was initiated by Harary and Schwenk in 1974 [1]. A graph G is called integral if all eigenvalues of its characteristic polynomial P(G,x) are integers. In general, the problem of finding (describing) all graphs with integral spectrum seems to be very difficult. Thus, it makes sense to restrict our investigations to interesting families of graphs. Particular results on integral graphs were obtained only for some certain classes of graphs [2]. Trees present another important family of graphs for which the problem has been considered in [1-13]. We know that trees of diameter 4 can be formed by joining the centers of r stars K 1,m1,K 1,m2,,K 1,mr to a new vertex v. The tree is denoted by S(r;m 1,,m r ) or simply S(r;m i ). Assume that the number of distinct integers of m 1,m 2,,m r is s. Without loss of generality, assume that the first s ones are the distinct integers such that 0 m 1 < m 2 < < m s. Suppose that a i is the multiplicity of m i for each i = 1,2,,s. The tree S(r;m i ) is also denoted by S(a 1 + a a s ;m 1,m 2,,m s ), where r = s a i and V = 1+ s a i (m i +1). For all other facts on graph spectra (or terminology), see [4]. In this paper, we investigate integral trees S(r;m i ) = S(a 1 +a 2 +a 3 ;m 1,m 2,m 3 ) of diameter 4. We shall construct infinitely many new classes of such integral trees by solving some certain Diophantine equations. These results are different from those in the existing literature. These results generalize some results of [10]. These integral graphs constructed in this paper are new members of integral graphs. We believe that it is useful for constructing other integral trees. 1 Preliminaries In this section, we state some known results on integral trees of diameter 4 and also obtain a new result on integral trees of diameter 4. Lemma 1.1 [7,10] For the tree S(r;m i ) = S(a 1 + +a s ;m 1,,m s ) of diameter 4, then we have P[S(r;m i ),x] =P[S(a a s ;m 1,,m s ),x] =x 1+ s a i (m i 1) [ (x 2 m i ) a i 1 (x 2 m i ) s a i j=1,j i ] (x 2 m j ).

3 2 Some New Families of Integral Trees of Diameter Four 3 Lemma 1.2 [10,13] The tree S(r;m i ) = S(a a s ;m 1,,m s ) of diameter 4 is integral if and only if (i) a i = 1 must hold if m i is not a perfect square, (ii) all solutions of the following equation are integers: (x 2 m i ) s a i j=1,j i (x 2 m j ) = 0. (1.1) We now discuss (1.1) to get more information. First, we divide both sides of (1.1) by s (x 2 m i ), and obtain F(x) = s a i x 2 m i 1 = 0. (1.2) Clearly, ± m i are not roots of (1.1) for 1 i s. Hence, all solutions of (1.1) are the same as those of (1.2). Because F(x) is an even function, we need only to consider the roots of F(x) on the real interval [0,+ ). Note that F(x) is discontinuous at each point m i. For 1 i s, when m i 0, we have that F( m i 0) =, F( m i + 0) = +, F(+ ) = 1. When m i = 0, we have that F(0 + 0) = +, and F (x) = 2x s 1 (x 2 m i. We deduce that F(x) is strictly monotone decreasing ) 2 on each of the continuous intervals over the real interval [0,+ ). Using Zero Point Theorem of mathematical analysis, we get that F(x) has s distinct positive real roots on the real interval [0,+ ). If 0 < u 1 < u 2 < < u s < + are the roots of F(x), then holds. holds. 0 m 1 < u 1 < m 2 < u 2 < < u s 1 < m s < u s < + (1.3) On the other hand, we note that (1.2) can be written as a 1 x 2 m 1 + a 2 x 2 m a s x 2 m s = 1. (1.4) From the above discussion and Lemma 1.2, we can deduce the following theorem Theorem 1.1 The tree S(r;m i )=S(a 1 + a a s ;m 1,m 2,,m s ) of diameter 4 is integral if and only if (i) a i = 1 must hold if m i is not a perfect square, (ii) all solutions of the following equation are integers: s a i x 2 m i = 1. (1.5)

4 4 WANG Ligong, ZHANG Zheng 15 ² Moreover, there exist positive integers u 1,u 2,,u s satisfying (1.3) such that the following linear equation system in a 1,a 2,,a s has positive integral solutions (a 1,a 2,,a s ), and such that a i = 1 must hold if m i is not a perfect square. a 1 a 2 u 2 1 m + 1 u 2 1 m u 2 1 m = 1, s. a 1 a 2 a s u 2 s m + 1 u 2 s m u 2 s m = 1. s a s (1.6) Theorem 1.2 [6,13] The tree S(r;m i )=S(a a s ;m 1,,m s ) of diameter 4 is integral if and only if there exist positive integers u i and nonnegative integers m i (i = 1,2,,s) such that 0 m 1 < u 1 < m 2 < u 2 < < u s 1 < m s < u s < +, and such that a k = (u 2 i m k),i k (m i m k ) (k = 1,2,,s) (1.7) are positive integers, and such that a i = 1 must hold if m i is not a perfect square. Theorem 1.3 [10] If the tree S(a 1 + a a s ;m 1,m 2,, m s ) of diameter 4 is integral, and m 1 ( 0),m 2 (> 0),,m s (> 0) are perfect squares, then for any positive integer n the tree S(a 1 n 2 + a 2 n 2 + +a s n 2 ;m 1 n 2,m 2 n 2,,m s n 2 ) is integral. Lemma 1.3 [10] For s = 3, integers m i ( 0),a i (> 0),u i (> 0) (i = 1,2,3) are given in Table 1. a i and u i (i = 1,2,3) are those of Theorem 1.2. Then for any positive integer n the tree S(a 1 n 2 + a 2 n 2 + +a 3 n 2 ;m 1 n 2,m 2 n 2,m 3 n 2 ) of diameter 4 is integral. Next we shall give a result on number theory. In the following symbol (a,b) = d denotes the greatest common divisor d of integers a, b, while a b (a b) means that a divides b (a does not divide b). Lemma 1.4 [14] Let a,b and c be integers with d = (a,b), we have (1) If d c, then the linear Diophantine equation in two variables ax + by = c (1.8) does not have integral solutions.

5 2 Some New Families of Integral Trees of Diameter Four 5 (2) If d c, then there are infinitely many integral solutions for (1.8). Moreover, if x = x 0,y = y 0 is a particular solution of (1.8), then all its solutions are given by x = x 0 + (b/d)t, y = y 0 (a/d)t, where t is an integer. Table 1 Integral trees S(a 1 n 2 + a 2 n 2 + a 3 n 2 ; m 1 n 2, m 2 n 2, m 3 n 2 ), where n 1. a 1 a 2 a 3 m 1 m 2 m 3 u 1 u 2 u Lemma 1.5 [15] Let m be a positive integer. If 2 m or 4 m, then there exist positive integral solutions for the Diophantine equation x 2 y 2 = m. (1.9) Remark 1.1 We can give a method for finding the solutions of (1.9). Suppose that m = m 1 m 2. Let x y = m 1, x + y = m 2 and 2 (m 1 + m 2 ). Then the solutions of (1.9) can be found easily (see [15]). 2 Integral trees of diameter four In this section, we shall construct infinitely many new classes of integral trees S(a 1 + a 2 + a 3 ;m 1,m 2, m 3 ) of diameter 4 from Lemma 1.3 and Theorem 1.2. They are different from those of [1-13].

6 6 WANG Ligong, ZHANG Zheng 15 ² The idea of constructing such integral tree is as follows: First, we properly choose integers m 1 ( 0), m 2 (> 0),,m s (> 0). Then, we try to find proper positive integers u i (i = 1,2,,s 1) satisfying (1.3) such that there are positive integral solutions (a 1,a 2,,a s ) for the linear equation system (1.6) (or such that all a k s of (1.7) are positive integers). Finally, we obtain positive integers a 1,a 2,,a s such that all the solutions a k s of (1.6) or (1.7) are integers. Thus, we have constructed many new classes of integral trees S(a a s ;m 1,,m s ) of diameter 4. From Lemma 1.3 and Theorem 1.2, we will construct infinitely many new classes of integral trees S(a 1 + a 2 + a 3 ;m 1,m 2,m 3 ) of diameter 4 and obtain the following theorem. Theorem 2.1 For s = 3, let integers m i ( 0), a i (> 0), u i (> 0) (i = 1,2,3) be those of Theorem 1.2, given in the following items, where t and k ( 0) are integers, then for any positive integer n the tree S(a 1 n 2 + a 2 n 2 + a 3 n 2 ;m 1 n 2,m 2 n 2,m 3 n 2 ) of diameter 4 is integral. (1) m 1 = 0, m 2 = 9, m 3 = 64, u 1 = 2, u 2 = 6, u 2 3 = 44t + 64 is a perfect square, a 1 = 11t + 16, a 2 = 12t + 15, a 3 = 21t. We have the following cases for (1), where k 1. (i) If t = k(11k 8), then u 2 3 = [2(11k 4)]2 is a perfect square, u 3 = 2(11k 4), a 1 = (11k 4) 2, a 2 = 3(2k 1)(22k 5), a 3 = 21k(11k 8). (ii) If t = k(11k + 8), then u 2 3 = [2(11k + 4)]2 is a perfect square, u 3 = 2(11k + 4), a 1 = (11k + 4) 2, a 2 = 3(2k + 1)(22k + 5), a 3 = 21k(11k + 8). (2) m 1 = 0, m 2 = 9, m 3 = 144, u 1 = 2, u 2 = 6, u 3 = 3k, a 1 = k 2, a 2 = k 2 1, a 3 = 7(k 4)(k + 4), where k 5. (3) m 1 = 0, m 2 = 25, m 3 = 81, u 1 = 2, u 2 = 6, u 2 3 = 225(8t + 1) is a perfect square, a 1 = 16(8t + 1), a 2 = 33(9t + 1), a 3 = 55(25t + 2). We have the following cases for (3), where k 1. (i) If t = k(2k 1), then u 2 3 = [15(4k 1)]2 is a perfect square, u 3 = 15(4k 1), a 1 = 16(4k 1) 2, a 2 = 33(6k 1)(3k 1), a 3 = 55(10k 1)(5k 2). (ii) If t = k(2k+1), then u 2 3 = [15(4k+1)]2 is a perfect square, u 3 = 15(4k+1), a 1 = 16(4k + 1) 2, a 2 = 33(6k + 1)(3k + 1), a 3 = 55(10k + 1)(5k + 2). (4) m 1 = 0, m 2 = 16, m 3 = 64, u 1 = 2, u 2 = 6, u 3 = 8k, a 1 = 9k 2, a 2 = 5(2k 1)(2k + 1), a 3 = 35(k 1)(k + 1), where k 2.

7 2 Some New Families of Integral Trees of Diameter Four 7 (5) m 1 = 0, m 2 = 25, m 3 = 256, u 1 = 2, u 2 = 6, u 3 = 20k, a 1 = 9k 2, a 2 = (4k 1)(4k + 1), a 3 = 15(5k 4)(5k + 4), where k 1. (6) m 1 = 0, m 2 = 16, m 3 = 144, u 1 = 2, u 2 = 6, u 2 3 = 16(8t + 1) is a perfect square, a 1 = 8t + 1, a 2 = 15t, a 3 = 105(t 1). We have the following cases for (6), where k 1. (i) If t = k(2k 1), then u 2 3 = [4(4k 1)]2 is a perfect square, u 3 = 4(4k 1), a 1 = (4k 1) 2, a 2 = 15k(2k 1), a 3 = 105(k 1)(2k + 1). (ii) If t = k(2k+1), then u 2 3 = [4(4k+1)]2 is a perfect square, u 3 = 4(4k+1), a 1 = (4k + 1) 2, a 2 = 15k(2k + 1), a 3 = 105(k + 1)(2k 1). (7) m 1 = 0, m 2 = 16, m 3 = 196, u 1 = 2, u 2 = 7, u 2 3 = 16(5t + 1) is a perfect square, a 1 = 5t + 1, a 2 = 11t, a 3 = 16(4t 9). We have the following cases for (7), where k 1. (i) If t = k(5k 2), then u 2 3 = [4(5k 1)]2 is a perfect square, u 3 = 4(5k 1), a 1 = (5k 1) 2, a 2 = 11k(5k 2), a 3 = 16(10k 9)(2k + 1). (ii) If t = k(5k+2), then u 2 3 = [4(5k+1)]2 is a perfect square, u 3 = 4(5k+1), a 1 = (5k + 1) 2, a 2 = 11k(5k + 2), a 3 = 16(10k + 9)(2k 1). (8) m 1 = 0, m 2 = 16, m 3 = 100, u 1 = 3, u 2 = 6, u 2 3 = 400(3t + 1) is a perfect square, a 1 = 81(3t + 1), a 2 = 5(25t + 8), a 3 = 208(4t + 1). We have the following cases for (8), where k 1. (i) If t = k(3k 2), then u 2 3 = [20(3k 1)]2 is a perfect square, u 3 = 20(3k 1), a 1 = 81(3k 1) 2, a 2 = 5(5k 2)(15k 4), a 3 = 208(2k 1)(6k 1). (ii) If t = k(3k+2), then u 2 3 = [20(3k+1)]2 is a perfect square, u 3 = 20(3k+1), a 1 = 81(3k + 1) 2, a 2 = 5(5k + 2)(15k + 4), a 3 = 208(2k + 1)(6k + 1). (9) m 1 = 0, m 2 = 36, m 3 = 144, u 1 = 3, u 2 = 8, u 3 = 6k, a 1 = 4k 2, a 2 = 7(k 1)(k + 1), a 3 = 25(k 2)(k + 2), where k 3. (10) m 1 = 0, m 2 = 16, m 3 = 144, u 1 = 3, u 2 = 8, u 2 3 = 16(8t + 1) is a perfect square, a 1 = 4(8t + 1), a 2 = 21t, a 3 = 75(t 1). We have the following cases for (10), where k 1. (i) If t = k(2k 1), then u 2 3 = [4(4k 1)]2 is a perfect square, u 3 = 4(4k 1), a 1 = 4(4k 1) 2, a 2 = 21k(2k 1), a 3 = 75(k 1)(2k + 1), where k 2.

8 8 WANG Ligong, ZHANG Zheng 15 ² (ii) If t = k(2k+1), then u 2 3 = [4(4k+1)]2 is a perfect square, u 3 = 4(4k+1), a 1 = 4(4k + 1) 2, a 2 = 21k(2k + 1), a 3 = 75(k + 1)(2k 1). (11) m 1 = 0, m 2 = 16, m 3 = 100, u 1 = 3, u 2 = 8, u 3 = 10k, a 1 = 36k 2, a 2 = (5k 2)(5k + 2), a 3 = 39(k 1)(k + 1), where k 2. (12) m 1 = 0, m 2 = 36, m 3 = 225, u 1 = 3, u 2 = 9, u 2 3 = 100(7t + 4) is a perfect square, a 1 = 9(7t + 4), a 2 = 5(25t + 13), a 3 = 128(4t + 1). We have the following cases for (12), where k 0. (i) If t = k(7k 4), then u 2 3 = [10(7k 2)]2 is a perfect square, u 3 = 10(7k 2), a 1 = 9(7k 2) 2, a 2 = 5(5k 1)(35k 13), a 3 = 128(2k 1)(14k 1), where k 1. (ii) If t = k(7k+4), then u 2 3 = [10(7k+2)]2 is a perfect square, u 3 = 10(7k+2), a 1 = 9(7k + 2) 2, a 2 = 5(5k + 1)(35k + 13), a 3 = 128(2k + 1)(14k + 1). (13) m 1 = 0, m 2 = 25, m 3 = 225, u 1 = 3, u 2 = 10, u 3 = 5k, a 1 = 4k 2, a 2 = 6(k 1)(k + 1), a 3 = 15(k 3)(k + 3), where k 4. (14) m 1 = 0, m 2 = 36, m 3 = 225, u 1 = 3, u 2 = 12, u 2 3 = 25(7t + 2) is a perfect square, a 1 = 4(7t + 2), a 2 = 3(25t + 2), a 3 = 72(t 1). We have the following cases for (14), where k 0. (i) If t = 7k 2 6k+1, then u 2 3 = [5(7k 3)]2 is a perfect square, u 3 = 5(7k 3), a 1 = 4(7k 3) 2, a 2 = 3(5k 3)(35k 9), a 3 = 72k(7k 6). (ii) If t = 7k 2 +6k+1, then u 2 3 = [5(7k+3)]2 is a perfect square, u 3 = 5(7k+3), a 1 = 4(7k + 3) 2, a 2 = 3(5k + 3)(35k + 9), a 3 = 72k(7k + 6). (iii) If t = 7k 2 8k+2, then u 2 3 = [5(7k 4)]2 is a perfect square, u 3 = 5(7k 4), a 1 = 4(7k 4) 2, a 2 = 3(5k 2)(35k 26), a 3 = 72(k 1)(7k 1). (iv) If t = 7k 2 +8k+2, then u 2 3 = [5(7k+4)]2 is a perfect square, u 3 = 5(7k+4), a 1 = 4(7k + 4) 2, a 2 = 3(5k + 2)(35k + 26), a 3 = 72(k + 1)(7k + 1). Proof We only prove (1), (2) (14) can be shown similarly to (1) by using Lemma 1.3 and Theorem 1.2. (1) For s = 3, by Lemma 1.3, we find m 1 = 0, m 2 = 9, m 3 = 64, u 1 = 2, u 2 = 6.

9 2 Some New Families of Integral Trees of Diameter Four 9 From Theorem 1.2, we get a 1 = (u2 1 m 1)(u 2 2 m 1)(u 2 3 m 1) (m 2 m 1 )(m 3 m 1 ) a 2 = (u2 1 m 2)(u 2 2 m 2)(u 2 3 m 2) (m 1 m 2 )(m 3 m 2 ) a 3 = (u2 1 m 3)(u 2 2 m 3)(u 2 3 m 3) (m 1 m 3 )(m 2 m 3 ) = u2 3 4, (2.1) = 3 11 (u2 3 9), (2.2) = (u2 3 64). (2.3) So, S(a 1 + a 2 + a 3 ;m 1,m 2,m 3 ) of diameter 4 is integral if and only if a 1, a 2, a 3 are positive integers. From (2.1) and (2.2), we get the Diophantine equation 12a 1 11a 2 = 27. (2.4) A result in elementary number theory (see also Lemma 1.4) yields that all positive integral solutions of (2.4) are given by a 1 = 11t + 16, a 2 = 12t + 15, where t( 1) is an integer. From (2.2) and (2.3), we have u 2 3 = 44t + 64, a 3 = 21t, where t is a nonnegative integer. Since u 2 3 (= 44t + 64) must be a perfect square, we deduce (u 3 + 8)(u 3 8) = 44t. (2.5) By Lemma 1.5 and Remark 1.1, we can obtain all positive integral solutions of (2.5) by discussing the following two cases for any positive integer k: Case 1 By (2.5), we assume that { u3 + 8 = 22k, u 3 8 = 2t/k. Then we get t = k(11k 8), u 3 = 2(11k 4). Hence, u 2 3 = [2(11k 4)]2 is a perfect square, a 1 = (11k 4) 2, a 2 = 3(2k 1)(22k 5), a 3 = 21k(11k 8). Case 2 By (2.5), we assume that { u3 8 = 22k, u = 2t/k. Then we get t = k(11k + 8), u 3 = 2(11k + 4). Hence, u 2 3 = [2(11k + 4)]2 is a perfect square, a 1 = (11k + 4) 2, a 2 = 3(2k + 1)(22k + 5), a 3 = 21k(11k + 8). Hence, when m 1 = 0, m 2 = 9, m 3 = 64, and a i (i = 1,2,3) are any case on the above two cases, then the tree S(a 1 + a 2 + a 3 ;m 1,m 2, m 3 ) of diameter 4 is integral. By Theorem 1.3, it is not difficult to prove that the tree S(a 1 n 2 + a 2 n 2 + a 3 n 2 ;m 1 n 2,m 2 n 2, m 3 n 2 ) of diameter 4 is also integral for any positive integer n.

10 10 WANG Ligong, ZHANG Zheng 15 ² References [1] Harary F, Schwenk A J. Which graphs have integral spectra?[m]. Graphs and Combinatorics, (eds. Bari R and Harary F), Lecture Notes in Mathematics 406, Berlin: Springer-Verlag, 1974, [2] Balińska K T, Cvetković D, Radosavljević Z, et al. A survey on integral graphs[j]. Univ Beograd, Publ Elektrotehn Fak Ser Mat, 2002, 13: [3] Cao Zhenfu. On the integral trees of diameter R when 3 R 6[J]. J Heilongjiang University, 1988, 2: 1-3, 95 (in Chinese). [4] Cvetković D, Doob M, Sachs H. Spectra of Graphs Theory and Application[M]. New York, Francisco, London: Academic Press, [5] Híc P, Nedela R. Balanced integral trees[j]. Math Slovaca, 1998, 48(5): [6] Li Maosheng, Yang Wensheng, Wang Jabao. Notes on the spectra of trees with small diameters[j]. J Changsha Railway University, 2000, 18(2): (in Chinese). [7] Li Xueliang, Lin Guoning. On trees with integer eigenvalues[j]. Chinese Science Bulletin, 1987, 32(11): (in Chinese). [8] Wang Ligong, Li Xueliang. Some new classes of integral trees with diameters 4 and 6[J]. Australasian J Combinatorics, 2000, 21: [9] Wang Ligong, Li Xueliang, Yao Xiangjuan. Integral trees with diameters 4, 6 and 8[J]. Australasian J Combinatorics, 2002, 25: [10] Wang Ligong, Li Xueliang, Zhang Shenggui. Families of integral trees with diameters 4, 6 and 8[J]. Discrete Appl Math, 2004, 136(2-3): [11] Watanabe M, Schwenk A J. Integral starlike trees[j]. J Austral Math Soc Ser A, 1979, 28: [12] Yuan Pingzhi. Integral trees of diameter 4[J]. J Systems Sci Math Sci, 1998, 18(2): (in Chinese). [13] Zhang Delong, Tan Shangwang. On integral trees of diameter 4[J]. J Systems Sci Math Sci, 2000, 20(3): (in Chinese). [14] Rosen K H. Elementary Number Theory and Its Applications[M]. London, Amsterdam: Addison-Wesley Publishing Company, 1984, [15] Pan Chengdong, Pan Chengbiao. Elementary Number Theory[M]. Beijing: Peking University Press, 1994 (in Chinese).

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

### The Open University s repository of research publications and other research outputs

Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

### Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Math 181 Handout 16. Rich Schwartz. March 9, 2010

Math 8 Handout 6 Rich Schwartz March 9, 200 The purpose of this handout is to describe continued fractions and their connection to hyperbolic geometry. The Gauss Map Given any x (0, ) we define γ(x) =

### LOW-DEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO

LOW-DEGREE PLANAR MONOMIALS IN CHARACTERISTIC TWO PETER MÜLLER AND MICHAEL E. ZIEVE Abstract. Planar functions over finite fields give rise to finite projective planes and other combinatorial objects.

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### arxiv:1203.1525v1 [math.co] 7 Mar 2012

Constructing subset partition graphs with strong adjacency and end-point count properties Nicolai Hähnle haehnle@math.tu-berlin.de arxiv:1203.1525v1 [math.co] 7 Mar 2012 March 8, 2012 Abstract Kim defined

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

### MATH 590: Meshfree Methods

MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### Solutions for Practice problems on proofs

Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some

### YILUN SHANG. e λi. i=1

LOWER BOUNDS FOR THE ESTRADA INDEX OF GRAPHS YILUN SHANG Abstract. Let G be a graph with n vertices and λ 1,λ,...,λ n be its eigenvalues. The Estrada index of G is defined as EE(G = n eλ i. In this paper,

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### Breaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and

Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study

### On Some Vertex Degree Based Graph Invariants

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan

### Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### 5. Factoring by the QF method

5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### On the number-theoretic functions ν(n) and Ω(n)

ACTA ARITHMETICA LXXVIII.1 (1996) On the number-theoretic functions ν(n) and Ω(n) by Jiahai Kan (Nanjing) 1. Introduction. Let d(n) denote the divisor function, ν(n) the number of distinct prime factors,

### FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT MINING SYSTEM

International Journal of Innovative Computing, Information and Control ICIC International c 0 ISSN 34-48 Volume 8, Number 8, August 0 pp. 4 FUZZY CLUSTERING ANALYSIS OF DATA MINING: APPLICATION TO AN ACCIDENT

### The Australian Journal of Mathematical Analysis and Applications

The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 1-14, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO

### Duplicating and its Applications in Batch Scheduling

Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China

### = 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### Factoring Dickson polynomials over finite fields

Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### Rolle s Theorem. q( x) = 1

Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question

### Determinants can be used to solve a linear system of equations using Cramer s Rule.

2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

### FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS

International Electronic Journal of Algebra Volume 6 (2009) 95-106 FACTORING CERTAIN INFINITE ABELIAN GROUPS BY DISTORTED CYCLIC SUBSETS Sándor Szabó Received: 11 November 2008; Revised: 13 March 2009

### Handout NUMBER THEORY

Handout of NUMBER THEORY by Kus Prihantoso Krisnawan MATHEMATICS DEPARTMENT FACULTY OF MATHEMATICS AND NATURAL SCIENCES YOGYAKARTA STATE UNIVERSITY 2012 Contents Contents i 1 Some Preliminary Considerations

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### An Overview of the Degree/Diameter Problem for Directed, Undirected and Mixed Graphs

Mirka Miller (invited speaker) University of Newcastle, Australia University of West Bohemia, Pilsen King s College, London Abstract A well-known fundamental problem in extremal graph theory is the degree/diameter

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### THE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:

THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces

### Classification of Cartan matrices

Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### z 0 and y even had the form

Gaussian Integers The concepts of divisibility, primality and factoring are actually more general than the discussion so far. For the moment, we have been working in the integers, which we denote by Z

### International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY

January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.

### On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems

Dynamics at the Horsetooth Volume 2, 2010. On using numerical algebraic geometry to find Lyapunov functions of polynomial dynamical systems Eric Hanson Department of Mathematics Colorado State University

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### The Method of Least Squares

The Method of Least Squares Steven J. Miller Mathematics Department Brown University Providence, RI 0292 Abstract The Method of Least Squares is a procedure to determine the best fit line to data; the

### ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

### On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

### Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1

Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing

### FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem

Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of fault-tolerant

### SOLUTIONS FOR PROBLEM SET 2

SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such

### Algebra 1 Course Title

Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

### Zeros of Polynomial Functions

Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

### ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath

International Electronic Journal of Algebra Volume 7 (2010) 140-151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

### On three zero-sum Ramsey-type problems

On three zero-sum Ramsey-type problems Noga Alon Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Yair Caro Department of Mathematics

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### ELA http://math.technion.ac.il/iic/ela

SIGN PATTERNS THAT ALLOW EVENTUAL POSITIVITY ABRAHAM BERMAN, MINERVA CATRAL, LUZ M. DEALBA, ABED ELHASHASH, FRANK J. HALL, LESLIE HOGBEN, IN-JAE KIM, D. D. OLESKY, PABLO TARAZAGA, MICHAEL J. TSATSOMEROS,

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Arithmetic and Algebra of Matrices

Arithmetic and Algebra of Matrices Math 572: Algebra for Middle School Teachers The University of Montana 1 The Real Numbers 2 Classroom Connection: Systems of Linear Equations 3 Rational Numbers 4 Irrational

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s13661-015-0508-0 R E S E A R C H Open Access Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

### Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Factoring Algorithms

Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### Factoring Cubic Polynomials

Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than

### Several Classes of Codes and Sequences Derived From a 4 -Valued Quadratic Form Nian Li, Xiaohu Tang, Member, IEEE, and Tor Helleseth, Fellow, IEEE

7618 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 57, NO 11, NOVEMBER 2011 Several Classes of Codes Sequences Derived From a 4 -Valued Quadratic Form Nian Li, Xiaohu Tang, Member, IEEE, Tor Helleseth,