Maximizing the order of a regular graph of given valency and second eigenvalue
Abstract
From Alon and Boppana, and Serre, we know that for any given integer and real number , there are only finitely many regular graphs whose second largest eigenvalue is at most . In this paper, we investigate the largest number of vertices of such graphs.
1 Introduction
For a regular graph on vertices, we denote by the eigenvalues of the adjacency matrix of . For a general reference on the eigenvalues of graphs, see [9, 18].
The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 9, 24] for example). In this paper, we investigate the maximum order of a connected regular graph whose second largest eigenvalue is at most some given parameter . As a consequence of work of Alon and Boppana, and of Serre [1, 12, 16, 24, 25, 28, 31, 35, 36, 42], we know that is finite for . The recent result of Marcus, Spielman and Srivastava [29] showing the existence of infinite families of Ramanujan graphs of any degree at least implies that is infinite for .
For any , the parameter 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 is not a complete graph, then contains an induced subgraph isomorphic to , so Cauchy eigenvalue interlacing (see [9, Proposition 3.2.1]) implies , contradiction. Thus for any and the unique graph meeting this bound is . The parameter can be determined using the fact that a graph with exactly one positive eigenvalue must be a complete multipartite graph (see [7, page 89]). The largest regular complete multipartite graph is the complete bipartite graph , since a regular partite graph has vertices. Thus , and is the unique graph meeting this bound. The values of and also follow from Theorem 2.3 in Section 2 below.
Results from Bussemaker, Cvetković and Seidel [10] and Cameron, Goethals, Seidel, and Shult [11] give a characterization of the regular graphs with smallest eigenvalue . Since the second eigenvalue of the complement of a regular graph is , the regular graphs with second eigenvalue are also characterized. This characterization can be used to find (see Section 3).
The values remaining to be investigated are for . The parameter has been studied by Teranishi and Yasuno [44] and Høholdt and Justesen [22] for the class of bipartite graphs in connection with problems in design theory, finite geometry and coding theory. Some results involving were obtained by Koledin and Staníc [26, 27, 43] and Richey, Shutty and Stover [47] who implemented Serre’s quantitative version of the Alon–Boppana Theorem [42] to obtain upper bounds for for several values of and . For certain values of and , Richey, Shutty and Stover [47] made some conjectures about . We will prove some of their conjectures and disprove others in this paper. Reingold, Vadhan and Wigderson [38] used regular graphs with small second eigenvalue as the starting point of their iterative construction of infinite families of expander using the zigzag product. Guo, Mohar, and TayfehRezaie [19, 32, 33] studied a similar problem involving the median eigenvalue. Nozaki [37] investigated a related, but different problem from the one studied 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 problems of minimizing for regular graphs of given order , valency and girth .
In this paper, we determine explicitly for several values of , confirming or disproving several conjectures in [47],
3  18  366  
4  50  4760  
5  9  804468  
6  16  16  
8  21  30  
10  114  32  
12  800  17  
4  39216  32  
6  10  34  
10  18  77  
14  24  18  
18  146  34  
30  1170  36  
126  74898  19  
5  11  36  
8  20  38  
9  27  614  
10  56  10440  
26  182  3017196  
35  1640  20  
80  132860  38  
728  12  40  
6  22  21  
10  24  40  
16  13  42  
42  24  762  
170  26  14480  
2730  266  5227320  
7  2928  22  
12  354312  42  
15  14  44  
62  26  23  
312  28  44  
7812  15  46  
8  28  100  
14  30 
and we find the graphs (in many cases unique) which meet our bounds. In many cases these graphs are distanceregular. For definitions and notations related to distanceregular graphs, we refer the reader to [9, Chapter 12]. Table 1 contains a summary of the values of that we found for . Table 2 contains six infinite families of graphs and seven sporadic graphs meeting the bound for some values of due to Theorem 2.3. Table 3 illustrates that the graphs in Table 2 that meet the bound also meet the bound for certain due to Proposition 2.9.
2 Linear programming method
In this section, we give a bound for using the linear programming method developed by Nozaki [37]. Let be orthogonal polynomials defined by the threeterm recurrence relation:
and
for . The following is called the linear programming bound for regular graphs.
Theorem 2.1 (Nozaki [37]).
Let be a connected regular graph with vertices. Let be the distinct eigenvalues of . Suppose there exists a polynomial such that , for any , , and for any . Then we have
Using Theorem 2.1, Nozaki [37] proved Theorem 2.2 below. Note that the paper [37] deals only with the problem of minimizing the second eigenvalue of a regular graph of given order and valency. While related to the problem of estimating , the problem considered by Nozaki in [37] is quite different from the one we study in this paper.
Theorem 2.2 (Nozaki [37]).
Let be a connected regular graph of girth , with vertices. Assume the number of distinct eigenvalues of is . If holds, then has the smallest secondlargest eigenvalue in all regular graphs with vertices.
Note also that while Table 2 is similar to [37, Table 2], the problems and tools in our paper are significantly different from the ones in [37].
Let be the tridiagonal matrix with lower diagonal , upper diagonal , and with constant row sum , where is a positive real number. Theorem 2.3 is the main theorem in this section and gives a new comprehension of the linear programming method and a general upper bound for without any assumption regarding the existence of some particular graphs.
Theorem 2.3.
If is the second largest eigenvalue of , then
(1) 
Let be a regular connected graph with second largest eigenvalue at most , valency , and vertices. Then if and only if is distanceregular with quotient matrix with respect to the distancepartition.
Proof.
We first show that the eigenvalues of that are not equal to , coincide with the zeros of (see also [7, Section 4.1 B]). Indeed,
and
by the threeterm recurrence relation, where . This equation implies that the zeros of are eigenvalues of . The monic polynomials form a sequence of orthogonal polynomials with respect to some positive weight on the interval [37]. Since the zeros of and interlace on , the zeros of are simple. Therefore all eigenvalues of coincide with the zeros of , and are simple.
Let be the eigenvalues of . We prove that the polynomial
(2) 
satisfies for . Note that it trivially holds that , and for any . The polynomial can be expressed as
(3) 
By [13, Proposition 3.2], has positive coefficients in terms of . This implies that has positive coefficients in terms of . Therefore for by [37, Theorem 3].
The polynomial can be expressed as . By [37, Theorem 3], we get that . Using Theorem 2.1 for , we obtain that
By [37, Remark 2], the graph attaining the bound has girth at least , and at most distinct eigenvalues. Therefore the graph is a distanceregular graph with quotient matrix by [37, Theorem 6] and [14]. Conversely the distanceregular graph with quotient matrix clearly attains the bound . ∎
Remark 2.4.
The distanceregular graphs which have as a quotient matrix of the distance partition are precisely the distanceregular graphs with intersection array .
Corollary 2.5.
Let be a connected regular graph with at least vertices. Let be the second largest eigenvalue of . Then holds with equality only if meets the bound .
Proof.
By Theorem 2.3, if , then the order of is at most . If the order of is equal to , then has at most distinct eigenvalues by [37, Remark 2]. However then the order of is less than by the Moore bound, a contradiction. Therefore if , then the order of is less than . Namely if the order of is at least , then . If holds, then the order of is bounded above by in Theorem 2.3, and attains the bound. ∎
We will discuss a possible second eigenvalue of . Indeed for any there exist such that is the second eigenvalue of . Let , be the largest zero of , , respectively. The zero can be expressed by , where [3, Section III.3].
Proposition 2.6.
The following hold:

for any , .

for and any , and , or and .

for and , or and .
Proof.
Since , we have for any , . Note that has a unique zero greater than . By the equality , we obtain that
This finishes the proof of the proposition. ∎
Remark 2.7.
The second largest eigenvalue of is the largest zero of . Since the zeros of and interlace, is a monotonically decreasing function in . In particular, , , and .
Note that both and form a sequence of orthogonal polynomials with respect to some positive weight on the interval . By Remark 2.7, the second eigenvalue of may equal all possible values between and . The following proposition shows that we may assume in Theorem 2.3 to obtain better bounds.
Proposition 2.8.
For any such that , there exist , such that both the secondlargest eigenvalues of and are . Then we have .
Proof.
Because , we get . Similarly . Note that and . Therefore
Table 2 shows the known examples attaining the bound . The incidence graphs of , , and are known to be unique for , , and , respectively (see, for example, [7, Table 6.5 and the following comments]). The incidence graphs of , , and are the Heawood graph, the TutteCoxeter graph (or Tutte 8cage), and the Tutte 12cage, respectively.
Graph meeting bound  Unique?  Ref.  

cycle  yes  
Complete graph  yes  
Complete bipartite graph  yes  
incidence graph of  ?  [7, 41]  
incidence graph of  ?  [4, 7]  
incidence graph of  ?  [4, 7]  
Petersen graph  yes  [21]  
Odd graph  yes  [34]  
Hoffman–Singleton graph  yes  [21]  
Clebsch graph  yes  [18, 40]  
Gewirtz graph  yes  [8, 17]  
graph  yes  [6, 20]  
Higman–Sims graph  yes  [17, 20] 
: projective plane, : generalized quadrangle,
: generalized hexagon, : prime power
The bounds in Table 2 solve several conjectures of Richey, Shutty, and Stover [47]. Richey, Shutty, and Stover prove that , but they note that the largest 3regular graph with they are aware of is the TutteCoxeter graph on 30 vertices. They conjectured that . They show that and conjecture that the largest 4regular graph with is the socalled rolling cube graph on 24 vertices (that is, the bipartite double of the cuboctahedral graph which is the line graph of the cube). They also conjectured that and the largest 4regular graph with is the Doyle graph on 27 vertices (see [15, 23] for a description of this graph). In Table 2 we confirm that and the TutteCoxeter graph (the incidence graph of ) is, in fact, the unique graph which meets this bound (see [7, Theorem 7.5.1] for uniqueness). However, Table 2 shows that (the Odd graph ) and that (the incidence graph of ), disproving the latter two conjectures.
Since the order of a graph is an integer, can be bounded above by . The graphs meeting the bound can be maximal under the assumption of a larger second eigenvalue.
Proposition 2.9.
Let , be the second largest eigenvalues of and , respectively. Suppose there exists a graph which attains the bound of Theorem 2.3. Then

If , then for . Moreover if is even, and is odd, then for .

If , for . Moreover if is even, and is odd, then for .
Proof.
We show only (1) because (2) can be proved similarly. For , we have
Therefore . If is odd, must be even. For , we have
Thus if is even, then . ∎
The larger second eigenvalues in Proposition 2.9 are calculated in Table 3. The graphs in Table 3 meet for any , where is the largest zero of in the table.
Graph  

(: even)  2  1  
(: odd)  2  1  
(: even)  3  
(: odd)  3  
(: even)  
(: odd)  
(: even)  5  
(: odd)  5  
(: even)  7  
(: odd)  7  
Petersen  3  1  1.11207  
Odd graph  4  2  2.02156  
Hoffman–Singleton  3  1  2.02845  
Clebsch  3  2  
Gewirtz  3  2  
3  4  
Higman–Sims  3  6 
is the largest zero of
By Theorem 2.3, we can obtain an alternative proof of the theorem due to Alon and Boppana, and Serre (see [1, 12, 16, 24, 25, 28, 31, 35, 36, 42] for more details).
Corollary 2.10 (Alon–Boppana, Serre).
For given , , there exist finitely many regular graphs whose second largest eigenvalue is at most .
Proof.
The second largest eigenvalue of is equal to the largest zero of . The zero is expressed by , where is less than [3, Section III.3]. This implies that there exists a sufficiently large such that . Therefore we have
3 Second largest eigenvalue
In this section, we classify the graphs meeting . The complement of a regular graph with second eigenvalue at most has smallest eigenvalue at least . The structure of such graph is obtained from a subset of a root system, and it is characterized as a line graph except for sporadic examples [7, Theorem 3.12.2]. The following theorem is immediate by [7, Theorem 3.12.2].
Theorem 3.1.
Let be a connected regular graph with vertices, valency , and second largest eigenvalue at most . Then one of the following holds:

is the complement of the line graph of a regular or a bipartite semiregular connected graph.

, and is a subgraph of the complement of , switchingequivalent to the line graph of a graph on eight vertices, where all valencies of have the same parity graphs nos. – in Table in [10].

, and is a subgraph of the complement of the Schlfli graph graphs nos. – in Table in [10].

, and is a subgraph of the complement of the Clebsch graph graphs nos. – in Table in [10].
The following theorem shows the classification of graphs meeting . Note that this result will show that for large whereas Theorem 2.3 would give a larger upper bound for .
Theorem 3.2.
Let be a connected regular graph with second largest eigenvalue at most , with vertices. Then the following hold:

, and is the hexagon.

, and is the Petersen graph.

, and is the complement of the graph no. in Table in [10].

, and is the Clebsch graph.

, and is the complement of the line graph of the complete graph with vertices, or the complement of one of the graphs nos. – in Table in [10].

, and is the complement of one of the graphs nos. – in Table in [10].

, and is the complement of one of the graphs nos. , in Table in [10].

, and is the complement of the graph no. in Table in [10].

, and is the complement of the Schlfli graph.

for , and is the complement of the line graph of .
Proof.
(1): A connected regular graph is an cycle, whose eigenvalues are (