SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1. For any λ<0, the parameter v(k, λ) can be determined using the fact that a graph with only one nonnegative eigenvalue is a complete graph. Indeed, if a graph has only one nonnegative eigenvalue, then it must be connected. If a connected graph G is not a complete graph, then G contains an induced subgraph isomorphic to K1,2,soCauchy eigenvalue interlacing (see [8, Proposition 3.2.1]) implies λ2(G) ≥ λ2(K1,2)=0,a contradiction. Thus v(k, λ)=k +1 foranyλ<0 and the unique graph meeting this bound is Kk+1. The parameter v(k, 0) can be determined using the fact that a graph with exactly one positive eigenvalue must be a complete multipartite graph (see [6, p. 89]). The largest k-regular complete multipartite graph is the complete bipartite graph Kk,k, since a k-regular t-partite graph has tk/(t − 1) vertices. Thus ∗Received by the editors July 15, 2015; accepted for publication May 16, 2016; published elec- tronically August 16, 2016. http://www.siam.org/journals/sidma/30-3/M103093.html †Department of Mathematical Sciences, University of Delaware, Newark, DE 19716-2553 (cioaba@ udel.edu). The research of this author was supported by the NSA grant H98230-13-1-0267 and NSF DMS-1600768. ‡School of Mathematical Sciences, University of Science and Technology of China, and Wen-Tsun Wu Key Laboratory of the Chinese Academy of Sciences, Hefei, Anhui, China ([email protected]). This author is partially supported by the National Natural Science Foundation of China (11471009). He also acknowledges the financial support of the Chinese Academy of Sciences under its “100 talent” program. §Department of Mathematics, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi 448-8542, Japan ([email protected]). This author is supported by JSPS KAKENHI grants 25800011, 26400003, and 16K17569. ¶Natural Sciences Division, Missouri Baptist University, Saint Louis, MO (Vermettej@mobap. edu). 1509 1510 CIOABA,ˇ KOOLEN, NOZAKI, AND VERMETTE v(k, 0) = 2k,andKk,k is the unique graph meeting this bound. The values of v(k, −1) and v(k, 0) also follow from Theorem 2.3 in section 2 below. Results from Bussemaker, Cvetkovi´c, and Seidel [9] and Cameron, Goethals, Sei- del, and Shult [10] give a characterization of the regular graphs with smallest eigen- value λmin ≥−2. Since the second eigenvalue of the complement of a regular graph is λ2 = −1 − λmin, the regular graphs with second eigenvalue λ2 ≤ 1 are also charac- terized. This characterization can be used to find v(k, 1) (see section√ 3). The values remaining to be investigated are v(k, λ)for1<λ<2 k − 1. The pa- rameter v(k, λ) has been studied by Teranishi and Yasuno [42] and Høholdt and Juste- sen [21] for the class of bipartite graphs in connection with problems in design theory, finite geometry, and coding theory. Some results involving v(k, λ) were obtained by Koledin and Stan´ıc [25, 26, 41] and Richey, Shutty, and Stover [45], who implemented Serre’s quantitative version of the Alon–Boppana theorem [40] to obtain upper bounds for v(k, λ) for several values of k and λ. For certain values of k and λ, Richey, Shutty, and Stover [45] made some conjectures about v(k, λ). We will prove some of their conjectures and disprove others in this paper. Reingold, Vadhan, and Wigderson [37] used regular graphs with small second eigenvalue as the starting point of their iter- ative construction of infinite families of expander using the zig-zag product. Guo, Mohar, and Tayfeh-Rezaie [18, 31, 32] studied a similar problem involving the median eigenvalue. Nozaki [36] investigated a related but different problem from the one stud- ied in our paper, namely, finding the regular graphs of given valency and order with smallest second eigenvalue. Amit, Hoory, and Linial [2] studied a related problem of minimizing max(|λ2|, |λn|) for regular graphs of given order n, valency k,andgirthg. In this paper, we determine v(k, λ) explicitly for several values of (k, λ), confirm- ing or disproving several conjectures in [45], and we find the graphs (in many cases unique) which meet our bounds. In many cases these graphs are distance-regular. For definitions and notation related to distance-regular graphs, we refer the reader to [8, Chapter 12]. Table 1 contains a summary of the values of v(k, λ) that we found for k ≤ 22. Table 2 contains six infinite families of graphs and seven sporadic graphs meeting the bound v(k, λ) for some values of k, λ due to Theorem 2.3. Table 3 illus- trates that the graphs in Table 2 that meet the bound v(k, λ) also meet the bound v(k, λ) for certain λ >λdue to Proposition 2.9. 2. Linear programming method. In this section, we give a bound for v(k, λ) (k) using the linear programming method developed by Nozaki [36]. Let Fi = Fi be orthogonal polynomials defined by the three-term recurrence relation: (k) (k) (k) 2 F0 (x)=1,F1 (x)=x, F2 (x)=x − k, and (k) (k) (k) Fi (x)=xFi−1(x) − (k − 1)Fi−2(x) for i ≥ 3. The following is called the linear programming bound for regular graphs. Theorem 2.1 (Nozaki [36]). Let G be a connected k-regular graph with v ver- tices. Let λ1 = k, λ2,...,λn be the distinct eigenvalues of G. Suppose there exists (k) a polynomial f(x)= i≥0 fiFi (x) such that f(k) > 0, f(λi) ≤ 0 for any i ≥ 2, f0 > 0,andfi ≥ 0 for any i ≥ 1. Then we have f(k) v ≤ . f0 LARGE GRAPHS OF GIVEN DEGREE AND SECOND EIGENVALUE 1511 Table 1 Summary of our results for k ≤ 22. (k, λ) v(k, λ) (k, λ) v(k, λ) (k, λ) v(k, λ) √ (2, −1) 3 (7, 1) 18 14, 13 366 √ (2, 0) 4 (7, 2) 50 14, 26 4760 √ √ , 1 − , − , 2 2 5 1 5 (8 1) 9 14 39 804468 , , , − (2√1) 6 (8 0) 16 (15 1) 16 2, 2 8 (8, 1) 21 (15, 0) 30 √ √ , 1 , , 2 2 5+1 10 8 7 114 (15 1) 32 √ √ 2, 3 12 8, 14 800 (16, −1) 17 √ (3, −1) 4 8, 21 39216 (16, 0) 32 (3, 0) 6 (9, −1) 10 (16, 1) 34 , , , (3√1) 10 (9 0) 18 (16 2) 77 3, 2 14 (9, 1) 24 (17, −1) 18 √ √ 3, 3 18 9, 2 2 146 (17, 0) 34 , , , (3√2) 30 (9 √4) 1170 (17 1) 36 3, 6 126 9, 2 6 74898 (18, −1) 19 (4, −1) 5 (10, −1) 11 (18, 0) 36 , , , (4 0) 8 (10 0) 20 (18√1) 38 (4, 1) 9 (10, 1) 27 18, 17 614 √ √ 4, 5 − 1 10 (10, 2) 56 18, 34 10440 √ √ 4, 3 26 (10, 3) 182 18, 51 3017196 √ (4, 2) 35 10, 3 2 1640 (19, −1) 20 √ √ 4, 6 80 10, 3 3 132860 (19, 0) 38 (4, 3) 728 (11, −1) 12 (19, 1) 40 (5, −1) 6 (11, 0) 22 (20, −1) 21 (5, 0) 10 (11, 1) 24 (20, 0) 40 , , − , (5 1) 16 (12 1) 13 (20√1) 42 (5, 2) 42 (12, 0) 24 20, 19 762 √ √ 5, 2 2 170 (12, 1) 26 20, 38 14480 √ √ √ 5, 2 3 2730 12, 11 266 20, 57 5227320 √ (6, −1) 7 12, 22 2928 (21, −1) 22 √ (6, 0) 12 12, 33 354312 (21, 0) 42 , , − , (6√1) 15 (13 1) 14 (21 1) 44 6, 5 62 (13, 0) 26 (22, −1) 23 √ 6, 10 312 (13, 1) 28 (22, 0) 44 √ 6, 15 7812 (14, −1) 15 (22, 1) 46 (7, −1) 8 (14, 0) 28 (22, 2) 100 (7, 0) 14 (14, 1) 30 Using Theorem 2.1, Nozaki [36] proved Theorem 2.2 below.