BearWorks MSU Graduate Theses Spring 2021 Proper Sum Graphs Austin Nicholas Beard Missouri State University, [email protected] As with any intellectual project, the content and views expressed in this thesis may be considered objectionable by some readers. However, this student-scholar’s work has been judged to have academic value by the student’s thesis committee members trained in the discipline. The content and views expressed in this thesis are those of the student-scholar and are not endorsed by Missouri State University, its Graduate College, or its employees. Follow this and additional works at: https://bearworks.missouristate.edu/theses Part of the Algebra Commons Recommended Citation Beard, Austin Nicholas, "Proper Sum Graphs" (2021). MSU Graduate Theses. 3613. https://bearworks.missouristate.edu/theses/3613 This article or document was made available through BearWorks, the institutional repository of Missouri State University. The work contained in it may be protected by copyright and require permission of the copyright holder for reuse or redistribution. For more information, please contact [email protected]. PROPER SUM GRAPHS A Master's Thesis Presented to The Graduate College of Missouri State University In Partial Fulfillment Of the Requirements for the Degree Master of Science, Mathematics By Austin Nicholas Beard May 2021 PROPER SUM GRAPHS Mathematics Missouri State University, May 2021 Master of Science Austin Nicholas Beard ABSTRACT The Proper Sum Graph of a commutative ring with identity has the prime ideals as ver- tices, with two ideals adjacent if their sum is a proper ideal. This thesis expands upon the research of Dhorajia. We will cover the groundwork to understanding the basics of these graphs, and gradually narrow our efforts into the minimal prime ideals of the ring. KEYWORDS: proper, sum, graph, ideal, maximal, minimal, comaximal ii PROPER SUM GRAPHS By Austin Nicholas Beard A Master's Thesis Submitted to the Graduate College Of Missouri State University In Partial Fulfillment of the Requirements Master of Science, Mathematics May 2021 Approved: Cameron Wickham, Ph.D., Thesis Committee Chair Les Reid, Ph.D., Committee Member Steven Senger, Ph.D., Committee Member Julie Masterson, Ph.D., Dean of the Graduate College In the interest of academic freedom and the principle of free speech, approval of this the- sis indicates the format is acceptable and meets the academic criteria for the discipline as determined by the faculty that constitute the thesis committee. The content and views expressed in this thesis are those of the student-scholar and are not endorsed by Missouri State University, its Graduate College, or its employees. iii ACKNOWLEDGEMENTS I would like to thank all of my professors from my time at Missouri State Univer- sity. In particular, I would like to thank Dr. Wickham, Dr. Reid, Dr. Stanojevic, Dr. Sun, and Dr. Senger. Without these individuals, I would not be half the mathematician or per- son I am today. I would also like to thank Dr. DuBois and Ms. Rost for inspiring an early love of math. Additionally, I would like to thank Dr. Snow for his guidance in all other areas. Lastly, I would like to thank the Lady Bears, my friends, and family for all their support and love during my years of study. I dedicate this thesis to my parents. iv TABLE OF CONTENTS 1. Introduction . Page 1 2. Graph Theory Background . Page 2 3. Ring Theory Background . Page 5 4. Proper Sum Graphs . Page 9 5. Conclusion . Page 19 v LIST OF FIGURES 1 Graph J ...................................... 2 2 Graph J' ...................................... 2 3 Graph of S = K1;6 ................................ 3 4 Proper Sum Graph of Z ............................. 9 C[x;y] 5 Graph of R = xy ................................ 13 6 Graph of Z6[y]................................... 13 vi INTRODUCTION Over the past 30 years, the study of graphs based on algebraic structures has been a growing area of research. It began with Beck creating the zero divisor graph of a ring R, denoted Γ(R). This is the graph whose vertices are the nonzero zero divisors of R and edges exist between vertices if their product is 0 in R. Inspired by the work of Mulay [6], Wickham and Spiroff developed the graph of equivalence classes of zero divisors of a ring R, which is created from the classes of zero divisors based on the annihilator ideals, rather than the individual zero divisors [8]. D.F. Anderson and Badawi also introduced the to- tal graph of R, denoted T (Γ(R)), as the graph with all elements of R as vertices, and for distinct x; y 2 R; the vertices x and y are adjacent if and only if x + y 2 Z(R) [1]. This can be combined with some work from Sharma and Bhatwadekar in 1995 [1]. This paper established another construction for a graph of a commutative ring R known as the a co- maximal graph. For this graph, vertices are still elements of the ring, but there is an edge between two vertices x and y in R if xR + yR = R. This idea would be followed up by Dhorajia in [3]. In this paper he defined the S-proper sum graph as a graph of commu- tative ring with identity that has the prime ideals as vertices, with two ideals adjacent if their sum is a proper ideal. We explore this graph further in this thesis. We begin with some background results. Then we improve upon some of the results in [3]. 1 GRAPH THEORY BACKGROUND The following are some graph theory basics and definitions, followed by a more fo- cused discussion on diameter, star graphs, and subgraphs. For more information on any object discussed, please refer to Diestel. A graph G = (V; E) is a pair of sets such that E ⊆ [V ]2. That is to say, the el- ements of our edge set, E, are 2-element subsets of our vertex set, V . A vertex is con- sidered to be incident to an edge if v 2 e, then we have an edge e at the vertex v. Now consider an isomorphism between graphs. Let G = (V; E) and G0 = (V 0;E0). Then we consider these two graphs to be isomorphic, written G ∼= G0 if there exists a bijection φ : V ! V 0 with xy 2 E () φ(x)φ(y) 2 E0 for all x; y 2 V . Consider Figure 1 and Figure 2. Figure 1: Graph J Figure 2: Graph J' It is readily apparent that J = (V = fA; B; C; D; Eg;E = fAB; BC; CD; DE; EA; DAg). Hence, we can say that A is incident to AB; EA; and DA. We can also tell that J 0 = (V = fA; B; C; Dg;E = fAB; BC; CD; DAg). We can also tell that no isomorphism exists be- tween J and J 0 as there are more vertices in J than in J 0. A path is the nonempty graph P = (V; E) where V = fx0; x1; : : : ; xkg and E = fx0x1; x1x2; : : : ; xk−1xkg, where each xi are distinct. The number of edges of a path is its 2 length. The distance in G of two vertices x; y, notated as dG(x; y), is the length of a short- est path from x to y in G, if no such path exists, then d(x; y) = 1. The diameter of G is the greatest distance between any two vertices in G, notated as diamG. Example 1. Consider again the graphs J and J'. In Figure 1, a path exists from E to B of length 4, EA ∗ AD ∗ DC ∗ CB. However, the distance between E and B is not 4, as a shorter path exists; namely EA ∗ AB, which has a length of 2. After completing this process with every other pair of vertices, we can find that the diameter of J is 2. It can be shown that the distance between any two points in J 0 is at most 2, so its diameter will also be 2. A r-partite (with r ≥ 2) graph is a graph G = (V; E) such that V admits a par- tition into r classes such that every edge has its ends in different classes. That is to say, vertices in the same class must not be adjacent. If r = 2 we call the graph bipartite. The complete r-partite graph is denoted by Kn1;:::nr . Graphs of the form K1;n are called star graphs. The vertex in the singleton partition class of K1;n is the center of the star, and all star graphs are bipartite. Consider again graph J 0. This graph is bipartite, as it can be partitioned into 2 classes, with the first being A; C and the second being B; D. How- ever, graph J is not bipartite. This can be seen in the triangle formed by vertices A; D; and E. As all three points share edges, they are all adjacent to each other. As such, the graph cannot be bipartite. Now consider Figure 3, which is a new graph that we will name S = K1;6. Figure 3: Graph of S = K1;6 This is an example of a star graph. Hence A is the center of the star. It is easy to 3 see this graph is bipartite, where A is in a class by itself, and all the other vertices are in a single class. It is important to note that while our example is finite, that is not always the case. It is possible to have the star graph K1;1. Consider G = (V; E) and G0 = (V 0;E0). If V 0 ⊆ V and E0 ⊆ E then G0 is a subgraph of G, this is written as G0 ⊆ G.