§1 Introduction
Our graph theoretic notation is standard (see [4, 5]). Let be an undirected graph with vertex set and edge set . Given , the distance equals the minimum length of a path between and if such a path exists or otherwise. If is connected, then all the distances between its vertices are finite and the diameter of is defined as the maximum of , where the maximum is taken over all pairs . The pairwise distance and the diameter of a connected graph can be calculated efficiently using breadth first search. The Moore or degreediameter problem is a classical problem in combinatorics (see [31]).
Problem 1.
Given and , what is the maximum order of a connected regular graph of diameter ?
There is a well known upper bound for known as the Moore bound which is obtained as follows. If is a connected regular graph of diameter , then for any given vertex in and any , the number of vertices at distance from is at most . Therefore,
(1) 
We denote by the right handside of the above inequality. When , it is straightforward to note that and the maximum is attained by the cycle on vertices. For , it is easy to see that and the maximum is attained by the complete graph on vertices.
For , a classical result of Hoffman and Singleton [20] gives that equals the Moore bound only when (attained by the cycle ), (the Petersen graph), (the HoffmanSingleton graph) or possibly . The existence of a regular graph with diameter on vertices is a well known open problem in this area (see [10, 24, 28]). For and , Damerell [11] and independently, Bannai and Ito [2] proved that there are no graphs attaining the Moore bound (1).
The adjacency matrix is the matrix whose th entry equals the number of edges between and . This matrix is a real symmetric matrix and if is simple (no loops nor multiple edges), then is a symmetric matrix. Let be a given integer. We will use the following family of orthogonal polynomials:
(2)  
(3) 
for any . Let . The polynomials form a sequence of orthogonal polynomials with respect to the positive weight
on the interval (see [23, Section 4]). The polynomials in are called Geronimus polynomials [18].
For any vertices and of and any nonnegative integer , the entry of the matrix equals the number of walks of length between and . A walk in is called nonbacktracking if for any and for any (when ). The following result goes back to Singleton [38].
Proposition 2 (Singleton [38]).
Let be a connected regular graph with adjacency matrix . For any vertices and of and any nonnegative integer , the entry of the matrix equals the number of nonbacktracking walks of length between and .
The eigenvalues of are real and we denote them by , where . Sometimes, to highlight the dependence of the eigenvalues on a particular graph, we will use for . When is regular and connected, it is known that and that . It is also known that and that is an eigenvalue if and only if the graph is bipartite. The smallest eigenvalue of a regular graph has been used to determine the independence number of various interesting graphs (see Godsil and Meagher [17]).
The properties of the eigenvalues of a regular graph were essential in the proofs of Hoffman and Singleton [20] as well as Damerell [11] and Bannai and Ito [2].
The spectral gap is an important parameter in spectral graph theory and is closely related to the connectivity [15] and expansion properties of the graph [19]. Informally, expanders are sparse graphs with large spectral gap. More precisely, a family of graphs is called a family of expanders if

there exists such that each is a connected regular graph for and the number of vertices of goes to infinity as goes to infinity,

there is a positive constant such that for any .
The first condition above explains the denomination of sparse used at the beginning of this paragraph. This condition implies that the number of edges in is linear in its number of vertices for every . This is best possible in order of magnitude for connected graphs.
The second condition is algebraic and is equivalent to a combinatorial condition that each is highly connected (meaning their expansion constants are bounded away from
) and also equivalent to the probability condition that a random walk on
converges quickly to its stationary distribution. We refer to [19] for the precise descriptions of these conditions.A natural question arising from the previous considerations is how large can the spectral gap be for an regular connected graph ? Since we are interested in situations where is fixed, this is equivalent to asking how small can be for a connected regular graph . This was answered by the AlonBoppana theorem.
Theorem 3.
Let be a natural number.

AlonBoppana 1986. If is a connected regular graph with vertices, then
(4) where is a constant and is a quantity that goes to as goes to infinity.

Asymptotic AlonBoppana Theorem. If is a sequence of connected regular graphs such that as . Then
(5)
To my knowledge, there is no paper written by Alon and Boppana which contains the theorem above. The first appearance of this result that I am aware of, is in 1986 in Alon’s paper [1] where it is stated that
R. Boppana and the present author showed that for every regular graph on vertices .
Note that the in [1] is the smallest positive eigenvalue of the Laplacian of and it equals . Therefore the statement above is equivalent to
(6) 
There is a similar result to the AlonBoppana theorem that is due to Serre [37]. The meaning of this theorem below is that large regular graphs tend to have a positive proportion of eigenvalues trying to be greater than .
Theorem 4 (Serre [37]).
For any , there exists such that any regular graph on vertices has at least eigenvalues that are at least .
These results motivated the definition of Ramanujan graphs that was introduced by Lubotzky, Phillips and Sarnak [27]. A connected regular graph is called Ramanujan if all its eigenvalues (with the exception of and perhaps , if is bipartite) have absolute value at most . Lubotzky, Phillips and Sarnak [27] and independently Margulis [30] constructed infinite families of regular Ramanujan graphs when is a prime. These constructions used results from algebra and number theory closely related to a conjecture of Ramanujan regarding the number of ways of writing a natural number as a sum of four squares of a certain kind (see [27, 30] and also, [12] for a more detailed description of these results). For the longest time, it was not known whether infinite families of regular Ramanujan graphs exist for any . Marcus, Spielman and Srivastava [29] obtained a breakthrough result by showing that there exist infinite families of bipartite regular Ramanujan graphs for any . Their method of interlacing polynomials has been fundamental to this proof and has found applications in other areas of mathematics as well.
§2 Spectral Moore theorems for general graphs
Throughout the years, several proofs of the AlonBoppana theorem have appeared (see Lubotzky, Phillips and Sarnak [27], Nilli [33], Kahale [25], Friedman [16], Feng and Li [14], Li and Solé [26], Nilli [34] and Mohar [32]). Theorem 4 was proved by Serre [37] with a nonelementary proof (see also [12]). The first elementary proofs appeared around the same time by Cioabă [6] and Nilli [34]. Richey, Stover and Shutty [36] worked to turn Serre’s proof into a quantitative theorem and asked the following natural question.
Problem 5.
Given an integer and , what is the maximum order of a regular graph with ?
These authors obtained several results involving . In this section, we describe our recent results related to the problem above and its bipartite and hypergraph versions. See [9, 8, 7] and the references therein for more details and other related problems. The method that is fundamental to all these results is due to Nozaki [35] who proved the linear programming bound for graphs.
Theorem 6 (Nozaki [35]).
Let be a connected regular graph with vertices and distinct eigenvalues . If there exists a polynomial such that , for any , , and for any , then
Nozaki used this result to study the following problem.
Problem 7.
Given integers , what is the regular graph on vertices that has the smallest among all regular graphs on vertices ?
While similar to it, this problem is quite different from Problem 5.
In [8], the authors used Nozaki’s LP bound for graphs to obtain the following general upper bound for .
Theorem 8 (Cioabă, Koolen, Nozaki and Vermette [8]).
Given integers and a nonnegative real number , let be the tridiagonal matrix:
If equals the second largest eigenvalue of the matrix , then
We sketch below the ideas of the proof of this theorem. For , denote
(7) 
where the s are the orthogonal polynomials defined in equations (2) and (3). The polynomials also form a family of orthogonal polynomials. They satisfy the following properties:
(8)  
(9) 
for .
The eigenvalues of the matrix are the roots of and are distinct (see [8, Theorem 2.3]). If we denote them by , then the polynomial satisfies for . It is a bit more involved to check the other conditions from Theorem 6 and we refer the reader to [8] for the details to see how one can apply Nozaki’s LP bound to and obtain that
To make things more clear, note the following result.
Proposition 9.
Let be an integer. For any , there exists an integer and a positive number such that is the second largest eigenvalue of the matrix .
Let denote the largest root of and denote the largest root of . Note that and
(10) 
Bannai and Ito [3, Section III.3] showed that , where . Because , one can show that for any (see [8, Prop 2.6] for other properties of these eigenvalues). From the remarks following Theorem 8, note that the second largest eigenvalue of equals the largest root of the polynomial . Because the roots of and interlace, one obtains that is a decreasing function in and takes values between and . Taking into the account what happens when , namely that , we obtain the following result.
Proposition 10.
For , takes any value in the interval .
Putting these things together, one deduces that can take any possible value between and . There are several infinite families for which the precise values have been determined in [8], but there are several open problems for relatively small values of and . For example, with an example of a regular graph with on vertices being the 2nd subconstituent of the HoffmanSingleton graph. Theorem 8 can give (see [7]). Also, we know that (Heawood graph), but we don’t know the exact value of for any . Lastly, has been determined for (equals with Pappus graph as an example attaining it) and (it is
with the odd graph
meeting it), but we don’t know it for .§3 AlonBoppana and Serre theorems
We point out the relevance of these results in the context of AlonBoppana and Serre theorems. A typical AlonBoppana result is of the form: if is a connected regular graph with diameter , then
(11) 
See Friedman [16, Corollary 3.6] for the inequality above or Nilli [34, Theorem 1] for a slightly weaker bound. The equivalent contrapositive formulation of inequality (11) is the following: if is a connected regular graph of diameter with , then . By Moore bound (1), this implies that
(12) 
Obviously, the best bound one can achieve here is obtained for that with . When applying Theorem 8, let be a real number such that . Given the properties of the largest roots of the polynomials (see Proposition 10 and the paragraph containing it), we must have that either or . If , then there exists such that is the largest eigenvalue of the polynomial . If is a connected regular graph with , then using Theorem 8 we get that
(13) 
which is clearly better than (12). If , then there exists such that is the largest eigenvalue of the polynomial . As above, if is a connected regular graph with , then Theorem 8 implies that
(14) 
which is again better than (11).
§4 Spectral Moore theorems for bipartite graphs
Building on this work, the author with Koolen and Nozaki extended and refined these results to bipartite regular graphs [9]. Let be an integer and be any real number between and . Define as the maximum number of vertices of a bipartite regular graph whose second largest eigenvalue is at most . Clearly, and a natural question is whether or not these parameters are actually the same or not. One can show that attained by the Petersen graph and that attained by the dimensional cube. For the bipartite graphs, a linear programming bound similar to Nozaki’s Theorem 6 from [35] was obtained with the use of the following polynomials:
for any , where were defined earlier in (2) and (3). Let be a bipartite connected regular graph. Its adjacency matrix has the form and we call the biadjacency matrix of . Note that for any . The following is called the LP bound for bipartite regular graphs.
Theorem 11 (Cioabă, Koolen and Nozaki [9]).
Let be a connected bipartite regular graph with vertices and denote by its set of distinct eigenvalues, where . If there exists a polynomial such that , for each , , and for each , then
(15) 
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 .
For any integers and any positive , let be the tridiagonal matrix with lower diagonal , upper diagonal , and constant row sum . Using Theorem 11, the following general upper bound for was obtained in [9].
Theorem 12 (Cioabă, Koolen and Nozaki [9]).
If is the second largest eigenvalue of , then
(16) 
Equality holds if and only if there exists a bipartite distanceregular graph whose quotient matrix with respect to the distancepartition from a vertex is for or for .
Define for . These are orthogonal polynomials and one can show that for as well as that . The first step in proving the above result is showing that the characteristic polynomial of equals . The proof proceeds in similar steps to Theorem 8, but is more technical and we refer the reader to [9] for the details. Similar to the situation for general graphs, one can show the following.
Proposition 13.
Let be an integer. For any , there exists and such that is the second largest eigenvalue of .
In [9], the authors also proved that for given and , the upper bound obtained in Theorem 12 is better than the one in Theorem 8. Theorem 12 has applications to various areas and it improves results obtained in the context of coding theory by Høholdt and Janwa [21] and Høholdt and Justensen [22] and design theory by Teranishi and Yasuno [39].
As in the case of Theorem 8, Theorem 12 has applications for the AlonBoppana theorems for bipartite regular graphs. Corollary 4.11 in [9] is a consequence of Theorem 12 and states that if is a bipartite regular graph of order greater than (the right handside in Theorem 12), then , where is the second largest eigenvalue of . Li and Solé [26, Theorems 3 and 5] proved that if is a bipartite regular graph of girth , then . This result follows from Corollary 4.11 in [9] as is the second largest eigenvalue of and having girth implies that has at least vertices.
§5 Classical Moore problem
Since the fundamental work of Singleton [38], Hoffman and Singleton [20], Bannai and Ito [2] and Damerell [11] in the 1970s, the families of orthogonal polynomials and have been important in the study of the Moore problem (1). It has been observed by several authors (see [13] or [31] for example) that if is connected regular of diameter , eigenvalues and , then
(17) 
where is the upper bound from the Moore bound (1). Recall that denotes the largest root of and satisfies
(18) 
Inequality (17) will improve the classical Moore bound (1) when . This will happen when . When , from (10) we know that . Note that [13, Theorem 2] contains a typo in the numerator of the right handside of the inequality (the in the numerator should be a ). The informal description of the result above is that when is large, then the order of the graph will be smaller than the Moore bound. Note however that if is small, then may be negative and inequality (17) may be worse than the classical Moore bound (1). Our results from [8] may be used to handle some cases when is small (actually when is small). More precisely, Theorem 8 gives an upper bound for the order of an regular graph with small second largest eigenvalue regardless of its diameter actually.
Known  Defect  Lower  Moore  Upper  

(8,2)  57  8  2.09503  2.19258  3.40512 
(9,2)  74  8  2.29956  2.37228  3.53113 
(10,2)  91  10  2.46923  2.54138  3.88473 
(4,3)  41  12  2.11232  2.25342  2.88396 
(5,3)  72  34  2.42905  2.62620  3.77862 
(4,4)  98  63  2.53756  2.69963  3.44307 
(5,4)  212  214  2.91829  3.12941  4.41922 
(3,5)  70  24  2.32340  2.39309  2.64401 
(4,5)  364  121  2.89153  2.93996  3.42069 
(3,6)  132  58  2.45777  2.51283  2.75001 
(4,6)  740  717  3.00233  3.08314  3.73149 
We explain this argument and give some numerical examples in Table 1 where we listed some values of (these values are from [31, page 4]) where the maximum orders of regular graphs of diameter are not known. For each such pair, the column labeled Known gives the largest known order of an regular graph of diameter . The column Defect equals the difference between the Moore bound and the entry in the Known column. The column Moore contains the value of rounded below to decimal points. The column Upper contains the lower bound for that guarantees that inequality (17) will give a lower bound than the value from the Known column. For example, for and , if is an regular graph with diameter having , then . The column Lower contains an upper bound for that guarantees that the order of such regular graph would be small. For example, for and , our Theorem 8 implies that if is an regular graph with , then . Another way to interpret these results in Table 1 is that if one wants to look for a regular graph of diameter with more than vertices, then the second largest eigenvalue of such putative graph has to be between and .
§6 Acknowledgments
This article is based on a talk I gave at the RIMS Conference Research on algebraic combinatorics, related groups and algebras. I thank the participants of the conference and I am grateful to Hiroshi Nozaki for his help with the computations in Table 1 and for his amazing support.
References
 [1] N. Alon, Eigenvalues and expanders, Combinatorica 6 (1986), 83–96.
 [2] E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 191–208.
 [3] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park CA 1984.
 [4] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer Universitext 2012.
 [5] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, 184, SpringerVerlag.
 [6] S.M. Cioabă, On the extreme eigenvalues of regular graphs, J. Combin. Theory Ser. B 96 (2006), 367–373.
 [7] S.M. Cioabă, J. Koolen, M. Mimura, H. Nozaki and T. Okada, Upper bounds for the order of a regular uniform hypergraph with prescribed eigenvalues, manuscript (2020).
 [8] S.M. Cioabă, J. Koolen, H. Nozaki and J. Vermette, Maximizing the order of a regular graph of given valency and second eigenvalue, SIAM J. Discrete Math. 30 (2016), 1509–1525.
 [9] S.M. Cioabă, J. Koolen and H. Nozaki, A spectral version of the Moore problem for bipartite regular graphs, Algebr. Comb. 2 (2019), 1219–1238.
 [10] C. Dalfó, A survey on the missing Moore graph, Linear Algebra Appl. 569 (2019), 1–14.
 [11] R.M. Damerell, On Moore graphs. Proc. Cambridge Philos. Soc. 74 (1973), 227–236.
 [12] G. Davidoff, P. Sarnak and A. Valette, Elementary number theory, group theory, and Ramanujan graphs, Cambridge University Press 2003.
 [13] M. Dinitz, M. Schapira and G. Shahaf, Approximate Moore graphs are good expanders, J. Combin. Theory Ser. B 141 (2020) 240–263.
 [14] K. Feng and W.C. Winnie Li, Spectra of hypergraphs and applications, J. Number Theory 60 (1996), no. 1, 1–22.
 [15] M. Fiedler, Algebraic connectivity of graphs, Czech. Math. J. 23 (1973), 298–305.

[16]
J. Friedman, Some geometric aspects of graphs and their eigenfunctions,
Duke Math. J. 69 (1993), 487–525.  [17] C. Godsil and K. Meagher, ErdősKoRado Theorems: Algebraic Approaches, Cambridge University Press, Cambridge, 2016.
 [18] J. Geronimus, On a set of polynomials, Ann. of Math. (2) 31 (1930), 681–686.
 [19] S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 4, 439–561.
 [20] A.J. Hoffman and R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. Res. Develop. 4 (1960), 497–504.
 [21] T. Høholdt and H. Janwa, Eigenvalues and expansion of bipartite graphs, Des. Codes Cryptogr. 65 (2012), 259–273.
 [22] T. Høholdt and J. Justesen, On the sizes of expander graphs and minimum distances of graph codes, Discrete Math. 325 (2014), 38–46.
 [23] A. Hora and N. Obata, Quantum probability and spectral analysis of graphs, Theoretical and Mathematical Physics, Springer, Berlin 2007.
 [24] A. Jurišić and J. Vidali, The Sylvester graph and Moore graphs, European J. Combin. 80 (2019), 184–193.
 [25] N. Kahale, On the second eigenvalue and linear expansion of regular graphs, Proc. 33rd IEEE FOCS, Pittsburgh, IEEE (1992), 296–303.
 [26] W.C. Winnie Li and P. Solé, Spectra of regular graphs and hypergraphs and orthogonal polynomials, European J. Combin. 17 (1996), 461–477.
 [27] A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–277.
 [28] M. Mačaj and J.Širáň, Search for properties of the missing Moore graph, Linear Algebra Appl. 432 (2010), no. 9, 2381–2398.
 [29] A. Marcus, D. Spielman and N. Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Ann. of Math. (2) 182 (2015), 307–325.
 [30] G.A. Margulis, Explicit grouptheoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problems of Information Transmission 24 (1988), 39–46.
 [31] M. Miller and J.Širáň, Moore graphs and beyond: A survey of the degree/diameter problem, Electron. J. Combin. (2005), DS14.
 [32] B. Mohar, A strengthening and a multipartite generalization of the AlonBoppanaSerre theorem, Proc. Amer. Math. Soc. 138 (2010), 3899–3909.
 [33] A. Nilli, On the second eigenvalue of a graph, Discrete Math. 91 (1991), 207–210.

[34]
A. Nilli, Tight estimates for eigenvalues of regular graphs,
Electron. J. Combin. 11 (2004), no. 1, Note 9, 4 pp.  [35] H. Nozaki, Linear programming bounds for regular graphs, Graphs Combin. 31 (2015), no. 6, 1973–1984.
 [36] J. Richey, N. Shutty, and M. Stover, Explicit bounds from the AlonBoppana theorem, Exp. Math. 27 (2018), no. 4, 444–453.
 [37] J.P. Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke , J. Amer. Math. Soc. 10 (1997), 75–102.
 [38] R. Singleton, On minimal graphs of maximum even girth, J. Combinatorial Theory 1 (1966), 306–332.
 [39] Y. Teranishi and F. Yasuno, The second largest eigenvalues of regular bipartite graphs, Kyushu J. Math. 54 (2000), 39–54.
Comments
There are no comments yet.