Let be a connected -regular simple graph with vertices. For , let denote the -th largest eigenvalue of the adjacency matrix of . The eigenvalues have close relationships with other graph invariants. The smallest eigenvalue is related to the diameter, the chromatic number and the independence number (see  or [8, Chapter 4] for example). The second eigenvalue plays a fundamental role in the study of expanders [2, 3, 8, 20]. Let denote the maximum order of a connected -regular graph with . For , from work of Alon and Boppana, and Serre, we know that the value is finite (see  and the references therein). In , we obtained the following upper bound for . Let be the tridiagonal matrix with lower diagonal , upper diagonal , and with constant row sum . If is the second largest eigenvalue of , then
Equality holds in (1.1) if and only if there is a distance-regular graph of valency with second largest eigenvalue , girth and diameter satisfying . For , there are no such graphs . However, for smaller values of , there are infinitely many values of and where the above inequality gives the exact value of .
In this paper, we improve the above results from  for bipartite regular graphs. Let denote the maximum order of a connected bipartite -regular graph with . Bipartite regular graphs with
have been classified for, , and . We obtain a general upper bound for for any . Our bound gives the exact value of whenever there exists a bipartite distance-regular graph of degree with second largest eigenvalue , diameter and girth such that . For certain values of , there are infinitely many such graphs of various valencies . When or , we prove the non-existence of bipartite distance-regular graphs with . Our results generalize previous work of Høholdt and Justesen  obtained in their study of graph codes and imply some results of Li and Solé  relating the second largest eigenvalue of a bipartite regular graph to its girth. The degree-diameter or Moore problem for graphs  is about determining the largest graphs of given maximum degree and diameter. Given the connections between the diameter and the second largest eigenvalue of bipartite regular graphs (see  for example), our Theorem 4.1 can be interpreted as a spectral version of the Moore problem for bipartite regular graphs.
In Section 2, we describe some sequences of orthogonal polynomials and develop the preliminary results and notation that will be used in the paper. In Section 3, we improve the linear programming bound from  for the class of bipartite regular graphs. In Section 4, we obtain the following upper bound for . Let be the tridiagonal matrix with lower diagonal , upper diagonal , and constant row sum . If is the second largest eigenvalue of , then
We show that equality happens in (1.2) when there is a bipartite distance-regular graph of degree , second largest eigenvalue having . Inequality (1.2) generalizes some results of Høholdt and Justesen  (see Corollaries 4.8 and 4.9), and of Li and Solé  (see Corollary 4.11). At the end of Section 4, we prove that the bound (1.2) is better than (1.1) for any and . In Section 5, we prove the non-existence of bipartite distance-regular graphs with for and . We conclude the paper with some remarks in Section 6.
In this section, we describe some useful polynomials that will be used to prove our main result. For any integer , let be a sequence of orthogonal polynomials defined by the three-term recurrence relation:
for . The notation is abbreviated to for the rest of the paper. Let . The polynomials form a sequence of orthogonal polynomials with respect to the positive weight
for . Note that for any , and
are even and odd functions of, respectively.
For , let and . It follows that for . By (2.2), the polynomials and satisfy the following properties:
for any if , and if . Note that for . For , the polynomials form a sequence of orthogonal polynomials with respect to the positive weight
on the interval .
For , let . A simple calculation implies that
Let denote the polynomial
It follows that . Using (2.4), the polynomial can be expressed as
for any if , and if . From Lemmas 3.3 and 3.5 in , for , the polynomials form a sequence of orthogonal polynomials with respect to the positive weight on the interval .
Let be the coefficients in . Then we have , and for any . Moreover if and only if .
We have that
By Theorem 3 in , we obtain that , and for any . Moreover if and only if . ∎
Let be a connected regular bipartite graph. The adjacency matrix of can be expressed by
where is the transpose matrix of . The matrix is called the biadjacency matrix of . It is not hard to see that
Since each entry of is non-negative , each entry of is also non-negative.
3 Linear programming bound for bipartite regular graphs
In this section, we give a linear programming bound for bipartite regular graphs. For general regular graphs, a linear programming bound was obtained by Nozaki .
Let be a connected bipartite -regular graph with vertices. Let be the set of distinct eigenvalues of , where . If there exists a polynomial such that , for each , , and for each , then
Equality holds if and only if for each , and for each , , and , where is the biadjacency matrix of . If equality holds and for each , then the girth of is at least .
From the spectral decomposition , we deduce that
is the identity matrix,, and is the all-ones matrix. Taking traces in both sides of (3.2), we get that
Therefore, . By using , we can obtain the same bound as (3.1).
If equality holds in (3.1), then for each , and for each , and . For the adjacency matrix , the -entry of is the number of non-backtracking walks of length from to . Since (2.7) and for each , there is no non-backtracking walk of length from to for each . Since is bipartite, the girth of is at least . ∎
4 Upper bound for bipartite graphs with given second eigenvalue
In this section, we obtain an upper bound on using the bipartite linear programming bound given by Theorem 3.1. Let be a real number and be an integer. Let be the tridiagonal matrix with lower diagonal , upper diagonal , and constant row sum . Let
If is the second largest eigenvalue of , then
Equality holds if and only if there exists a bipartite distance-regular graph whose quotient matrix with respect to the distance-partition from a vertex is for or for .
We first calculate the characteristic polynomial of . The polynomials , , , and are defined in Section 2. Note that is the characteristic polynomial of the principal matrix formed by the first rows and columns of for . By this fact and equations (2.2) and (2.4), we can compute
Since the zeros of and interlace on , each zero of is simple and belongs to except for the smallest zero. For the smallest zero is equal to because by (2.3) and (2.5). For , the smallest zero is negative. From , each non-zero real eigenvalue of has multiplicity 1, and if , then has imaginary eigenvalues.
Let be the polynomial
We show that satisfies the condition of the linear programming bound from Theorem 3.1 for bipartite graphs. Note that , and for each . It suffices to show that for each .
The polynomial can be expressed by
where if is even, and if is odd. Thus,
The polynomial can be expressed by . By Lemma 2.1, we have
By applying Theorem 3.1 to the polynomial , we have
By Theorem 3.1, the bipartite graph attaining the bound has girth at least , and at most distinct eigenvalues. Since the diameter is at most , the graph satisfies , where is the girth and is the diameter. By , the graph becomes a distance-regular graph [1, Theorem 4.4], , and it must have the quotient matrix for , or for (see Proposition 4.6 below). Conversely the distance-regular graph with the quotient matrix clearly attains the bound . ∎
Note that is a distance-regular graph with the quotient matrix if and only if is a connected bipartite -regular graph that has only distinct eigenvalues, and whose girth is at least . Table 1 shows the known examples attaining the bound [7, Section 6.11].
|Complete bipartite graph|
|4||minus a parallel class|
: affine plane, : generalized quadrangle, : generalized hexagon,
: partial geometry, : prime power, : power of 2,
We use the bipartite incidence graph of an incidence structure.
Recall that denotes the maximum order of a connected (not necessarily bipartite) -regular graph whose second largest eigenvalue is at most . We have , which is attained by the Petersen graph  and from Table 1, which is attained by the bipartite incidence graph of the symmetric -design.
The following is the bipartite version of Theorem 5 in .
Let be a bipartite distance-regular graph of order with quotient matrix with respect to the distance-partition from a vertex. Then for any bipartite -regular graph of order .
Assume that there exists a graph of order such that . Then also attains the bound from Theorem 4.1. This implies that must have the eigenvalue , which is a contradiction. ∎
Let (resp. ) denote the largest zero of (resp. ).
For each , there exist such that is the second largest eigenvalue of .
Note that for because for . The second eigenvalue of is equal to the largest zero of . Since the zeros of and interlace, is a monotonically decreasing function in . In particular, with , , and . The largest zero of can be expressed by , where [4, Section III.3]. For , it follows from that . This implies that the possible value is between and . Therefore for each , there exist , such that is the second eigenvalue of . ∎
Note that for , is the second eigenvalue of both and for some , with , . By the following proposition, we may assume in Theorem 4.1 to obtain better bounds.
Let . Suppose and satisfy that , and the second largest eigenvalues of and are . Then we have .
Since holds, we have
Similarly holds. By , we have and . It therefore follows that
For , is the second eigenvalue of both and for some , with , . It follows that