scholarly journals A survey on integral graphs

2002 ◽  
pp. 42-65 ◽  
Author(s):  
Krystyna Balinska ◽  
Dragos Cvetkovic ◽  
Zoran Radosavljevic ◽  
Slobodan Simic ◽  
Dragan Stevanovic
Keyword(s):  
Integral Graph ◽  
Proof Techniques ◽  
Integral Graphs

A graph whose spectrum consists entirely of integers is called an integral graph. We present a survey of results on integral graphs and on the corresponding proof techniques.

scholarly journals Some Algebraic Properties of a Class of Integral Graphs Determined by Their Spectrum

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Jia-Bao Liu ◽  
S. Morteza Mirafzal ◽  
Ali Zafari
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Adjacency Matrix ◽  
Prime Integer ◽  
Integral Graph ◽  
Algebraic Properties ◽  
Integral Graphs

Let Γ = V , E be a graph. If all the eigenvalues of the adjacency matrix of the graph Γ are integers, then we say that Γ is an integral graph. A graph Γ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph Γ = Cay ℤ n , S , where n = p m ( p is a prime integer and m ∈ ℕ ) and S = a ∈ ℤ n | a , n = 1 . First, we show that Γ is an integral graph. Also, we determine the automorphism group of Γ . Moreover, we show that Γ and K v ▽ Γ are determined by their spectrum.

INTRODUCING PROOF TECHNIQUES USING THE LOGICAL GAMEMINE HUNTER

PRIMUS ◽  
1995 ◽  
Vol 5 (2) ◽  
pp. 108-112 ◽  
Author(s):  
Allan Alexander Struthers
Keyword(s):  
Proof Techniques

PARITY RESULTS FOR PARTITIONS WHEREIN EACH PART APPEARS AN ODD NUMBER OF TIMES

2018 ◽  
Vol 99 (1) ◽  
pp. 51-55
Author(s):  
MICHAEL D. HIRSCHHORN ◽  
JAMES A. SELLERS
Keyword(s):  
Generating Function ◽  
Proof Techniques ◽  
One To One

We consider the function $f(n)$ that enumerates partitions of weight $n$ wherein each part appears an odd number of times. Chern [‘Unlimited parity alternating partitions’, Quaest. Math. (to appear)] noted that such partitions can be placed in one-to-one correspondence with the partitions of $n$ which he calls unlimited parity alternating partitions with smallest part odd. Our goal is to study the parity of $f(n)$ in detail. In particular, we prove a characterisation of $f(2n)$ modulo 2 which implies that there are infinitely many Ramanujan-like congruences modulo 2 satisfied by the function $f.$ The proof techniques are elementary and involve classical generating function dissection tools.

Proof Techniques

Proofs 101 ◽  
2020 ◽  
pp. 15-36
Author(s):  
Joseph Kirtland
Keyword(s):  
Proof Techniques

scholarly journals Discrete Mathematics into K-11 and K-12 Grade Education

2021 ◽  
Vol 23 (05) ◽  
pp. 319-324
Author(s):  
Mr. Balaji. N ◽  
◽  
Dr. Karthik Pai B H ◽  
Keyword(s):  
Set Theory ◽  
Mathematical Reasoning ◽  
Mathematical Induction ◽  
Discrete Mathematics ◽  
College Classrooms ◽  
Underlying Principle ◽  
Education Study ◽  
Strongly Connected ◽  
Proof Techniques ◽  
K 12

Discrete mathematics is one of the significant part of K-11 and K-12 grade college classrooms. In this contribution, we discuss the usefulness of basic elementary, some of the intermediate discrete mathematics for K-11 and K-12 grade colleges. Then we formulate the targets and objectives of this education study. We introduced the discrete mathematics topics such as set theory and their representation, relations, functions, mathematical induction and proof techniques, counting and its underlying principle, probability and its theory and mathematical reasoning. Core of this contribution is proof techniques, counting and mathematical reasoning. Since all these three concepts of discrete mathematics is strongly connected and creates greater impact on students. Moreover, it is potentially useful in their life also out of the college study. We explain the importance, applications in computer science and the comments regarding introduction of such topics in discrete mathematics. Last part of this article provides the theoretical knowledge and practical usability will strengthen the made them understand easily.

Construction of integral graphs

2000 ◽  
Vol 15 (3) ◽  
pp. 239-246 ◽  
Author(s):  
Wang Ligong ◽  
Li Xueliang ◽  
Zhang Shenggui
Keyword(s):  
Integral Graphs

scholarly journals Which non-regular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues?

2004 ◽  
Vol 286 (1-2) ◽  
pp. 15-24 ◽  
Author(s):  
Krystyna T. Balińska ◽  
Slobodan K. Simić ◽  
Krzysztof T. Zwierzyński
Keyword(s):  
Maximum Degree ◽  
Integral Graphs
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