On bipartite cages of excess 4

The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $ n-M(k,g) $. In this paper we consider the existence of $(k,g)$-bipartite graphs of excess $4$ via studying spectral properties of their adjacency matrices. We prove that the $(k,g)$-bipartite graphs of excess $4$ satisfy the equation $kJ=(A+kI)(H_{d-1}(A)+E)$, where $A$ denotes the adjacency matrix of the graph in question, $J$ the $n \times n$ all-ones matrix, $E$ the adjacency matrix of a union of vertex-disjoint cycles, and $H_{d-1}(x)$ is the Dickson polynomial of the second kind with parameter $k-1$ and of degree $d-1$. We observe that the eigenvalues other than $\pm k$ of these graphs are roots of the polynomials $H_{d-1}(x)+\lambda$, where $\lambda$ is an eigenvalue of $E$. Based on the irreducibility of $H_{d-1}(x)\pm2$ we give necessary conditions for the existence of these graphs. If $E$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs \emph{graphs with cyclic excess}; if $E$ is the adjacency matrix of a disjoint union of two cycles we call the corresponding graphs \emph{graphs with bicyclic excess}. In this paper we prove the non-existence of $(k,g)$-graphs with cyclic excess $4$ if $k\geq6$ and $k \equiv1 \!\! \pmod {3}$, $g=8, 12, 16$ or $k \equiv2 \!\! \pmod {3}$, $g=8,$ and the non-existence of $(k,g)$-graphs with bicyclic excess $4$ if $k\geq7$ is odd number and $g=2d$ such that $d\geq4$ is even.


Introduction
A k-regular graph of girth g is called a (k, g)-graph. A (k, g)-cage is a (k, g)-graph with the fewest possible number of vertices, among all (k, g)-graphs. The order of a (k, g)-cage is denoted by n(k, g). The Cage Problem calls for finding cages, and this problem was considered for the first time by Tutte [16]. It is known that a (k, g)-graph exists for any combination of k ≥ 2 and g ≥ 3, [7,14]. However, the orders n(k, g) of (k, g)-cages have only been determined for very limited sets of parameters [9]. A natural lower bound on the order of a (k, g)-graph is called the Moore bound, and the form of the bound depends on the parity of g, i.e., n(k, g) ≥ M(k, g) = 1 + k + k(k − 1) + ... + k(k − 1) (g−3)/2 , g odd, 2 1 + (k − 1) + ... + (k − 1) (g−2)/2 , g even.
The graphs whose orders are equal to the Moore bound are called Moore graphs. They are known to exist if k = 2 and g ≥ 3, g = 3 and k ≥ 2, g = 4 and k ≥ 2, g = 5 and k = 2, 3, 7, or g = 6, 8, 12 and a generalized n-gon of order k − 1 exists [1,4,9]. The existence of a (57, 5)-Moore graph is an open question.
The excess e of a (k, g)-graph is the difference between its order n and the Moore bound M(k, g), i.e., e = n − M(k, g). Regarding graphs of even girth we will use the following three results: Theorem 1.1 ( [3]) Let G be a (k, g)-cage of girth g = 2d ≥ 6 and excess e. If e ≤ k − 2, then e is even and G is bipartite of diameter d + 1.
For the next theorem, let D(k, 2) denote the incidence graph of a symmetric (v, k, 2)design.
Motivated by the result in Theorem 1.3, which was obtained through counting cycles in a hypothetical graph with given parameters and excess 4, in this paper we address the question of the existence of (k, g)-graphs of excess 4 using spectral properties of their adjacency matrices. The question of the existence of (k, g)-graphs of excess 4 is wide open, and prior to the publication of [11], no such results were known. The results contained in our paper further extend our understanding of the structure of the potential graphs of excess 4. Throughout, we assume that k ≥ 6, g = 2d ≥ 6 and G is a (k, g)-graph of excess 4 and order n. Due to Biggs's result stated in Theorem 1.1, the restriction of the parameters k, g given above allows us to conclude that G is a bipartite graph with diameter d + 1. For each integer i in the range 0 ≤ i ≤ d + 1, we define the n × n matrix A i = A i (G) as follows. The rows and columns of A i correspond to the vertices of G, and the entry in position (u, v) is 1 if the distance d(u, v) between the vertices u and v is i, and zero otherwise. Clearly, A 0 = I, A 1 = A, the usual adjacency matrix of G. The last non-zero matrix is the matrix A d+1 which we shall denote by E and refer to it as the excess matrix i.e., E is the adjacency matrix of the graph with the same vertex set V as G such that two vertices of V are adjacent if and only if they have distance d + 1. We will call this graph the excess graph of G and we will denote it by G(E). If J is the all-ones matrix, the sum of the i-distance matrices To apply the last identity we will use Lemma 4 from [11]. Employing the methodology used by Bannai et al. in [1], [2], later by Biggs et al. in [3], Delorme et al. in [5] and Garbe in [10], we will show that the eigenvalues of G other than ±k are the roots of the polynomials H d−1 (x) + λ. Here, H d−1 (x) is the Dickson polynomial of the second kind with parameter k − 1 and degree d − 1, and λ is an eigenvalue of the excess matrix E. Furthermore, for odd k ≥ 7 and d ≥ 4, we prove that the polynomial H d−1 (x) ± 2 is irreducible over Q[x], which leads to necessary conditions for existence of (k, g)-graphs of excess 4, Theorem 2.7.
We say that a graph G has a cyclic excess if the excess graph G(E) is a cycle of length n, and a graph G has a bicyclic excess if G(E) is a disjoint union of two cycles. In [6] Delorme et al. considered graphs with cyclic defect and excess 2, proving nonexistence of infinitely many such graphs. The paper describes the cycle structure of the excess graphs of the known non-trivial graphs of excess 2: 1) the excess graph of the only (3, 5)-graph of excess 2 is a disjoint union of a 9-cycle and a 3-cycle or a disjoint union of an 8-cycle and 4-cycle; 2) the excess graph of the unique (4, 5)-graph of excess 2 (the Robertson graph) is a disjoint union of a 3-cycle, a 12-cycle and a 4-cycle; 3) the excess graph of the unique (3, 7)-graph of excess 2 (the McGee graph) is a disjoint union of six 4-cycles.
We note that no (k, g)-graph of cyclic excess 2 are known, while examples of graphs with bicyclic excess 2 can be found among the (3, 5)-graphs of excess 2. Proving that the excess graphs of bipartite graphs of excess 4 form a disjoint union of cycles, while also inspired by the results in [6], in Section 3 we consider the existence of bipartite graphs of excess 4 with cyclic and bicyclic excess 4. Based on the irreducibility of , we prove the non-existence of infinitely many such graphs of girths at least 8.
2 Necessary conditions for the existence of graphs of even girth and excess 4 Let k ≥ 6, g = 2d ≥ 6, and let G be a (k, g)-graph of excess 4. Then G is bipartite of diameter d + 1. Let N G (u, i) denote the set of vertices of G whose distance from u in G is equal to i, 1 ≤ i ≤ d + 1. The subgraph of G induced by the set of vertices of G whose distance from u is at most g−2 2 and whose distance from v is by one larger than their distance from u induces a tree of depth g−2 2 rooted at u (we will call it T u ). Also, the subgraph of G induced by the set of vertices of G whose distance from v is at most g−2 2 and whose distance from u is by one larger than their distance from v induces a tree of depth g−2 2 rooted at v (we will call it T v ). Since G is of girth g, the trees T u and T v are disjoint and contain no cycles. Since each vertex of G is of degree ). We will call the union of the trees T u , T v with the edge f Moore tree of G rooted at f ; it is the subtree of G that is the basis of the Moore bound for even g. The graph G must contain 4 additional vertices w 1 , w 2 , w 3 , w 4 which do not belong to either T u or T v , and whose distance from both u and v is greater than g−2 2 . We will call these vertices the excess vertices with respect to f and denote this set X f = {w 1 , w 2 , w 3 , w 4 }; we call the edges not contained in the Moore tree of G horizontal edges.
The following lemma restricts the possible ways in which the four excess vertices are attached to the Moore tree.
Lemma 2.1 ( [11]) Let k ≥ 6, g = 2d ≥ 6. Let G be a (k, g)-graph of excess 4, u, v be two adjacent vertices in G, and X f = {w 1 , w 2 , w 3 , w 4 } be the four excess vertices with respect to the edge f = {u, v}. The induced subgraph G[w 1 , w 2 , w 3 , w 4 ] is isomorphic to 2K 2 (two disjoint copies of K 2 ) or P 3 (a path of length 3).
Next, let us define the following polynomials: In [15], Singleton gives many relationships between these polynomials. We will use two of them. Given any i ≥ 0, The above defined polynomials have a close connection to the properties of a graph G. Namely, for t < g the element (F t (A)) x,y counts the number of paths of length t joining vertices x and y of G. It follows from (3) that G t (A) counts the number of paths of length at most t joining pairs of vertices in G. All of the preceding claims can be found in [5]. The next lemma is based on the structure of G described in Lemma 2.1: Proof. Let f = {u, v} be a base edge of the Moore tree and let We consider the case when G[w 1 , w 2 , w 3 , w 4 ] is isomorphic to 2K 2 in which case the excess vertices do not share common neighbour. The other cases when G[w 1 , w 2 , w 3 , w 4 ] is isomorphic to 2K 2 and the excess vertices share common neighbour or the subgraph induced by the excess vertices contains P 3 are analogous. Since there are k − 1 paths of length d from u to w 1 and w 3 , by the definition of Considering the vertices of distance d from u, there are also the (k − 1) d−1 leaves of the subtree rooted at v. Proof. By the definition of the polynomials G i (x) and using the fact that G has diameter d + 1 we conclude . Substituting this identity in (4), where we fix The next theorem gives a relationship between the eigenvalues of the matrices A and E (this result is an analogue of Theorem 3.1 in [5]): where λ is an eigenvalue of E.
Proof. Let us suppose that µ is an eigenvalue of A. Since G is a k-regular graph, the all-ones matrix J is a polynomial in A. This implies that any eigenvector of A is also an eigenvector of J. From kJ = (A + kI)(H d−1 (A) + E) and since H d−1 (A) is also a polynomial in A, we have that E is a polynomial in A, and consequently, every eigenvector of A is an eigenvector of E. Therefore, the eigenvalues of kJ are of the form (µ + k)(H d−1 (µ) + λ). As is well-known, the eigenvalues of kJ are kn (with multiplicity 1) and 0 (with multiplicity n − 1). The eigenvalue kn corresponds to µ = k, and so all the remaining eigenvalues, except for −k, satisfy the above equation.
q.e.d. Since the eigenvalues of a disjoint union of cycles are known, we are now in a position to determine the spectrum of A: Lemma 2.5 Let k ≥ 6, g = 2d ≥ 6 and let G be a (k, g)-graph of excess 4. If A is the adjacency matrix of G and E is the excess matrix of G, then: 1) The matrix E is the adjacency matrix of a graph G(E), consisting of a disjoint union of c cycles C i of length l i with 1 ≤ i ≤ c. Moreover, if d is odd and V 1 and V 2 are the two partition sets of the bipartite graph G, then every cycle in G(E) is completely contained either in V 1 or V 2 .
2) The spectrum of A consists of: Proof. 1) Our proof is analogous to that of Kovács for girth 5, [12], and Garbe's proof for odd girth g = 2k + 1 > 5, [10]. Let f = {u, v} be a base edge of a bipartite Moore tree of G. Lemma 2.1 asserts that there exist exactly two vertices of G on distance d + 1 from u. Namely, they are the excess vertices adjacent to the leaves of the subtree rooted at v. The excess matrix E is the adjacency matrix for the graph G(E) with same vertex set V as G such that two vertices of G(E) are adjacent if and only if they are of distance d + 1. Because for each vertex u ∈ V (G) there are exactly two vertices on distance d + 1 from u, every component of G(E) is a cycle. Let c be the number of these cycles and let l i , i = 1, .., c, be the lengths of these cycles ordered in an arbitrary manner. Moreover, if d is an odd number, any two vertices of G with distance d + 1 lie in the same partite set. Therefore any connected component of G(E) is entirely contained either in V 1 or V 2 .
2) The eigenvalues of an n-cycle are known and are equal to 2 cos( 2πj n ), (j = 0, ..., n − 1). Therefore the eigenvalues of G(E) are 2 cos( 2πj l i ), j = 0, 1, ..., l i − 1; 1 ≤ i ≤ c, [10]. Since G is a k-regular bipartite graph, it has (among others) the eigenvalues k and −k. Let V 1 and V 2 be the partition sets of G. Hence the eigenvector of A corresponding to k consist of the all-ones vector j, and the eigenvector corresponding to −k is the vector j ′ with values 1 on V 1 and values −1 on V 2 . If d is an odd number then two vertices of G(E) are adjacent if and only if they are in the same partite set. Therefore E · j ′ = 2j ′ , which implies that from the set of c solutions on H d−1 (x) = −2 we need to subtract two multiplicities for the eigenvalues k and −k of G. If d is an even number then two vertices of G(E) are adjacent if and only if they are in different partite sets. Thus E · j ′ = −2j ′ . In this case, from the set of c solutions on H d−1 (x) = −2 we need to subtract one multiplicity for the eigenvalue k and from the set of all solutions on H d−1 (x) = 2 we need to subtract one multiplicity for the eigenvalue −k. q.e.d.
Lemma 2.6 Let k ≥ 6, g = 2d ≥ 6 and let G be a (k, g)-graph of excess 4. Furthermore, let c be the number of cycles of G(E) and c 2 be the number of cycles of even length. Then: Proof. 1) Combining Theorem 2.4 and part 2) from Lemma 2.5 we obtain that H d−1 (x) − 2 is an irreducible factor of the characteristic polynomial of A. Realizing that the roots of an irreducible factor of a characteristic polynomial of given rational symmetric matrix have the same multiplicities, [12], from 2) of Lemma 2. q.e.d.
We can base the testing of irreducibility of H d−1 (x) ± 2 on the well-known Eisenstein's criterion that asserts for a polynomial f (x) = n i=0 a i x i ∈ Z[x] and a prime p that divides a i for all 0 ≤ i < n, does not divide a n and p 2 does not divide a 0 Now we are ready for the main result in this section: Proof. According to Lemma 2.6, it is enough to prove that the polynomials H d−1 (x)− 2 and H d−1 (x) + 2 are irreducible. We will prove using induction on d ≥ 4 that x. Let us suppose that the above formula holds for H d−2 (x) and H d−3 (x). That yields By the inductional hypothesis, follows that for an odd d occurs H d−1 (0) = (−1) and H d−1 (0) = 0 for an even d. Hence for an odd d ≥ 5 the absolute value (−1) ± 2 is not divisible by 2 2 , and clearly for an even d ≥ 4, ±2 is not divisible by 2 2 . Since k − 1 is even, it follows that every coefficient on H d−1 (x) ± 2 except for the coefficient 1 of x d−1 is divisible by 2. Thus, the conditions of the Eisenstein's criterion are satisfied, and H d−1 (x) ± 2 is irreducible. q.e.d. 3 The non-existence of bipartite graphs of cyclic or bicyclic excess In this section we still deal with the family of graphs considered as in Section 2. Again, let k ≥ 6, g = 2d ≥ 6 and let G be a (k, g)-graph of excess 4 and order n. Clearly n is even number. We have already proved that the excess graph G(E) consists of a disjoint union of c cycles C i , 1 ≤ i ≤ c. If c = 1 and G(E) consists of an n-cycle, G is of cyclic excess 4, and if c = 2 and G(E) consists of a disjoint union of two cycles, G is of bicyclic excess 4. These are the graphs we study in this section. Note that there are no graphs G with cyclic excess 4 if d is an odd number; in this case we showed that each cycle of G(E) is completely contained either in V 1 or V 2 .
Let d be an even number and let L n be an n-cycle formed by the vertices of G(E). If A ′ is the adjacency matrix of L n , its characteristic polynomial χ(L n , x) satisfies χ(L n , x) = (x − 2)(x + 2)(R n (x)) 2 , where R n is a monic polynomial of degree n 2 − 1. Consider the factorization x n − 1 = l|n Φ l (x), where Φ l (x) denotes the l-th cyclotomic polynomial. In the following paragraph, we summarize the properties of cyclotomic polynomials as listed in [6]. The cyclotomic polynomial Φ l (x) has integral coefficients, it is irreducible over Q[x], and it is self-reciprocal (x φ(l) Φ l (1/x) = Φ l (x)). From the irreducibility and the selfreciprocity of Φ l (x) follows that the degree of Φ l (x) is even for l ≥ 2. Thus we obtain the following factorization of R n (x) : Lemma 3.1 Let g = 2d > 6 and l ≥ 3 be a divisor of n. If there is a (k, g)-graph with cyclic excess 4 and order n, then F l,k,d−1 (x) must be reducible over , then all its roots must be eigenvalues of A. Employing Observation 3.1. from [6], we conclude that there are at most φ(l) roots of F l,k,d−1 (x) that are eigenvalues of A. Thus (d−1)· φ(l) 2 =φ(l) i.e., d = 3. This contradicts the assumption that 2d > 6. q.e.d.  Proof. Follows directly from Lemma 3.1, with the additional assumption f 3 (x) = x + 1, f 4 (x) = x and f 6 (x) = x − 1.
q.e.d. If n ≡ 0 (mod 4), then using the formula for the order of G, d − 1 must be odd. On the other hand, since we see that if d − 1 is an odd number then x divides H d−1 (x), which implies that H d−1 (x) is reducible. Therefore the second condition from Lemma 3.2 is satisfied. The irreducibility of the polynomials H d−1 (x) − 1 over Q[x] is examined in [5], where it is analytically proven that these polynomials are irreducible for d ∈ {4, 6, 8} and the paper contains a conjecture that d ≥ 10, H d−1 (x) − 1 is irreducible. From the irreducibility of H d−1 (x) − 1 we obtain the main non-existence result of our paper. Theorem 3.3 If k and g satisfy one of the following conditions, there exist no (k, g)graphs of cyclic excess 4: 1) k ≡ 1, 2 (mod 3) and g = 8; 2) k ≡ 1 (mod 3) and g = 12; 3) k ≡ 1 (mod 3) and g = 16.
q.e.d. Remark: Since d is an even number, Theorem 2.7 asserts that d − 1 divides c − 1 and c 2 − 1. This claim is satisfied because c = c 2 = 1.
Next, let us consider graphs of bicyclic excess 4. In this case, we can assume an arbitrary (even or odd) d, as this case does not depend of the parity of d. So, let G(E) be a graph consisting of a disjoint union of two cycles C 1 and C 2 . If d is an odd number, then the vertex sets of the cycles C 1 and C 2 correspond to the partite sets V 1 and V 2 , respectively. If n ≡ 0 (mod 4), d is an even, each edge of C(E) has endpoints in V 1 and V 2 , and therefore each of the cycles has even length, c 2 = 2. Furthermore, k − 1 must be odd. Unfortunately, this will not help us in excluding any family of pairs (k, g) for which G does not exist. In fact, for an odd d − 1 and an odd k − 1 we cannot conclude irreducibility of H d−1 (x) + 2, thus, we cannot employ Lemma 2.6. If n ≡ 2 (mod 4) and d is odd, then the lengths of C 1 and C 2 are equal to n 2 (clearly n = 2s + 1 is odd). Therefore c 2 = 0 and clearly d − 1 divides c − 2 and c 2 .
The main result about the non-existence of graphs G with bicyclic excess 4 is given in the following theorem: Theorem 3.4 If k ≥ 7 is an odd and g = 2d ≥ 8, where d is an even integer, then there exist no (k, g)-graphs with bicyclic excess 4.