Maximizing the order of a regular graph of given valency and second eigenvalue
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.
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  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  and Cameron, Goethals, Seidel, and Shult  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  and Høholdt and Justesen  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  who implemented Serre’s quantitative version of the Alon–Boppana Theorem  to obtain upper bounds for for several values of and . For certain values of and , Richey, Shutty and Stover  made some conjectures about . We will prove some of their conjectures and disprove others in this paper. Reingold, Vadhan and Wigderson  used regular graphs with small second eigenvalue as the starting point of their iterative construction of infinite families of expander using the zig-zag product. Guo, Mohar, and Tayfeh-Rezaie [19, 32, 33] studied a similar problem involving the median eigenvalue. Nozaki  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  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 ,
and we find the graphs (in many cases unique) which meet our bounds. In many cases these graphs are distance-regular. For definitions and notations related to distance-regular 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 . Let be orthogonal polynomials defined by the three-term recurrence relation:
for . The following is called the linear programming bound for regular graphs.
Theorem 2.1 (Nozaki ).
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  proved Theorem 2.2 below. Note that the paper  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  is quite different from the one we study in this paper.
Theorem 2.2 (Nozaki ).
Let be a connected -regular graph of girth , with vertices. Assume the number of distinct eigenvalues of is . If holds, then has the smallest second-largest eigenvalue in all -regular graphs with vertices.
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.
If is the second largest eigenvalue of , then
Let be a -regular connected graph with second largest eigenvalue at most , valency , and vertices. Then if and only if is distance-regular with quotient matrix with respect to the distance-partition.
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,
by the three-term 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 . 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
satisfies for . Note that it trivially holds that , and for any . The polynomial can be expressed as
By [37, Remark 2], the graph attaining the bound has girth at least , and at most distinct eigenvalues. Therefore the graph is a distance-regular graph with quotient matrix by [37, Theorem 6] and . Conversely the distance-regular graph with quotient matrix clearly attains the bound . ∎
The distance-regular graphs which have as a quotient matrix of the distance partition are precisely the distance-regular graphs with intersection array .
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 .
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].
The following hold:
for any , .
for and any , and , or and .
for and , or and .
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. ∎
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.
For any such that , there exist , such that both the second-largest eigenvalues of and are . Then we have .
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 Tutte-Coxeter graph (or Tutte 8-cage), and the Tutte 12-cage, respectively.
|Graph meeting bound||Unique?||Ref.|
|Complete bipartite graph||yes|
|incidence graph of||?||[7, 41]|
|incidence graph of||?||[4, 7]|
|incidence graph of||?||[4, 7]|
|Clebsch graph||yes||[18, 40]|
|Gewirtz graph||yes||[8, 17]|
|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 . Richey, Shutty, and Stover prove that , but they note that the largest 3-regular graph with they are aware of is the Tutte-Coxeter graph on 30 vertices. They conjectured that . They show that and conjecture that the largest 4-regular graph with is the so-called 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 4-regular 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 Tutte-Coxeter 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.
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 .
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 . ∎
is the largest zero of
Corollary 2.10 (Alon–Boppana, Serre).
For given , , there exist finitely many -regular graphs whose second largest eigenvalue is at most .
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].
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 , switching-equivalent to the line graph of a graph on eight vertices, where all valencies of have the same parity graphs nos. – in Table in .
, and is a subgraph of the complement of the Schlfli graph graphs nos. – in Table in .
, and is a subgraph of the complement of the Clebsch graph graphs nos. – in Table in .
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 .
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 .
, 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 .
, and is the complement of one of the graphs nos. – in Table in .
, and is the complement of one of the graphs nos. , in Table in .
, and is the complement of the graph no. in Table in .
, and is the complement of the Schlfli graph.
for , and is the complement of the line graph of .
(1): A connected -regular graph is an -cycle, whose eigenvalues are (