Spectral Threshold for Extremal Cyclic Edge-Connectivity

SPECTRAL THRESHOLD FOR EXTREMAL CYCLIC EDGE-CONNECTIVITY arXiv:2003.02393v1 [math.CO] 5 Mar 2020 SINAN G. AKSOY, MARK KEMPTON, AND STEPHEN J. YOUNG Abstract. The cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges whose removal disconnects G into components where every component contains a cycle. We show the cyclic edge-connectivity is defined for graphs with minimum degree at least 3 and girth at least 4, as long as G is not Kt,3 . From the proof of this result it follows that, other than K3,3 , the cyclic edge-connectivity is bounded above by (d − 2)g for d-regular graphs of girth g ≥ 4. We then provide a spectral condition for when this upper bound on cyclic edge-connectivity is tight. 1. Introduction The traditional notion of graph edge-connectivity is the smallest k such that there exists a set of edges S ⊆ E(G) with |S| = k, where G \ S is disconnected. Note that traditional edge-connectivity does not stipulate any conditions on properties of the components of G\S. The notion of conditional edge-connectivity, introduced by Harary [9], extends the traditional edge-connectivity by stipulating that components of G \ S satisfy some given property. More precisely: Definition 1 (Harary 1983 [9]). Let P be any property of a graph G = (V, E), and let S ⊂ E(G). The universal P -connectivity is the minimum |S| such that G \ S is disconnected, and every component of G \ S has property P . We note there are several formulations of conditional connectivity; for instance, the qualifier universal reflects that every component of G \ S has P , whereas existential conditional connectivity relaxes this condition to some component satisfying P . As discussed in [9], Harary introduced conditional connectivity with explicit hopes of providing a framework for devising connectivity concepts that are meaningful in applications. Indeed, Harary notes that conditional connectivity for various properties has naturally arisen in areas such as computer network reliability [5], VLSI and separator problems [10], among others. In this work, we consider the universal P -edge-connectivity, where P is the property of containing a cycle. We call this the cyclic edge-connectivity of G. Prior to Harary’s work, Bollobas alludes to cyclic edgeconnectivity [6, p. 113], and Harary proved the universal cyclic edge-connectivity of the balanced complete bipartite graph Kn,n is n2 − 2n for n even. The cycle condition is also natural for a number of applications, such as network reliability, as the existence of a cycle is necessary to guarantee multiple paths between pairs of vertices. Prior work has established a number of spectral bounds for traditional edge and vertex connectivity; see [1] and the references contained therein. However, spectral bounds on conditional connectivity appear far more rare and, to our knowledge, no such results exist for cyclic edge-connectivity. In Section 3 we will show that if a graph G is d-regular and has girth g, then the cyclic edge-connectivity is bounded above by (d − 2)g. Furthermore, we will provide a spectral condition for when this upper bound is tight. 2. Main tools and notation For a graph G = (V, E) and vertex subsets X, Y ⊆ V , let E(X, Y ) denote the set of edges between X and Y , and let e(X, Y ) := |E(X, Y )|. Further, let G[X] denote the subgraph induced by X. A primary tool we will use is the discrepancy inequality, also sometimes called the expander mixing lemma. Date: March 6, 2020. This work was supported by the High Performance Data Analytics (HPDA) program at Pacific Northwest National Laboratory. Pacific Northwest National Laboratory is operated by Battelle Memorial Institute under Contract DE-ACO6-76RL01830. PNNL Information Release: PNNL-SA-151831. 1 Discrepancy Inequality (Alon and Chung [2]). Let G = (V, E) be a d-regular, n-vertex graph with second largest adjacency eigenvalue λ. For all X, Y ⊆ V , s    d|X||Y | |X| |Y | ≤ λ |X||Y | 1 − e(X, Y ) − 1− . n n n The other key ingredient of the proof is the following theorem of Alon, Hoory, and Linial which provides a lower bound for the number of vertices in a graph with a given average degree and girth. This result may be thought of as irregular generalization of the result of Moore (see [4, p. 180]), which upper bounds the number of vertices in a d-regular graph of a given diameter. Irregular Moore Bound (Alon, Hoory, Linial [3]). The number of vertices n in a graph of girth g and average degree at least d ≥ 2 satisfies  r−1 X    1 + d (d − 1)i if g = 2r + 1   i=0 n0 (d, g) = . r−1 X   i  2 (d − 1) if g = 2r  i=0 To demonstrate the applicability of these tools, we first obtain the following naı̈ve spectral bound on conditional edge-connectivity in terms of girth when conditioned on the size of the minimal component. Proposition 1. Let γ be the size (number of edges) of the smallest edge-cut of a d-regular graph which results in components of size at least k, and let the girth be 2r. Then   k . γ ≥ k(d − λ) 1 − (d − 1)r − 2 Proof. We first note that by the minimality of the cut, there is some set X such that γ = e(X, X). Thus, by the Discrepancy Inequality, we have that s    d|X|(n − |X|) |X| n − |X| |X|(n − |X|) γ = e(X, X̄) ≥ − λ |X|(n − |X|) 1 − 1− = (d − λ) . n n n n Applying the Ireregular Moore bound of Alon [3] immediately yields the desired result.  The form of this lower bound is quite close to the trivial upper bound on γ, k(d − 2) + 2, stemming from the case where G[X] is a k-vertex tree. In some sense, this extremal example makes cyclic edge-connectivity the natural strengthening of the size-limited connectivity. It is also worth mentioning that Proposition 1 can be easily extended to the case of odd girth with a slightly more complicated spectral bound. The final tool we will utilize to study the cyclic edge-connectivity of d-regular graphs is the ear decomposition of a graph, which as stated in [7], may defined as follows. Ear Decomposition. For a subgraph F of G, an ear of F in G is a nontrivial path in G whose ends lie in F but whose internal vertices don’t. An ear decomposition of a 2-edge-connected graph G is a nested sequence (G0 , G1 , . . . , Gk ) of subgraphs of G such that (i) G0 is a cycle, (ii) Gi+1 = Gi ∪ Pi , where Pi is an ear of Gi in G, for 0 ≤ i ≤ k, (iii) Gk = G. 3. Cyclic Edge-Connectivity Before proceeding with our main result, we first address the existence of cyclic-edge connectivity. Indeed, for acyclic or unicyclic graphs, it is clear the cyclic edge connectivity does not exist. The work of Lovász [11] and Dirac [8] provide a complete characterization of the class of graphs with no pairs of vertex disjoint cycles. Roughly speaking, these are graphs obtained from K5 , a wheel, and K3,t plus any subset of edges connecting vertices in the three element class, and a forest plus a dominating vertex by the duplication and subdivision of edges and the addition of trees. However, as no assurances are provided on the relative sizes of 2 these cycles, these works do not yield an upper bound on the cyclic edge-connectivity even in the d-regular case. In Lemma 1, we show that in addition to cyclic edge-connectivity existing, there exists a valid edge cut in which one of the components is a minimum length cycle. Thus, if in addition the graph is regular with degree d, there is a trivial upper bound on the cyclic edge-connectivity of (d − 2)g by considering the edge cut induced by a length g cycle. Lemma 1. Let G be a graph with minimum degree d ≥ 3 and girth g ≥ 4. If G is not K3,t , then there exists a cycle C of length g such that every component of G − E(C, C) contains a cycle. Proof. We first consider the case that g ≥ 5 and let C be a cycle of length g. As C is an induced cycle it is clear that G[C] contains a cycle. Now consider an arbitrary component G[X] of G − E(C, C) and let x ∈ X. As  g g ≥ 5, the vertex x has at most one neighbor in C as otherwise there exists a cycle of length at most 2 + 1 < g. But then the minimum degree of G[X] is 2 and hence it contains a cycle. Now suppose that G is a graph with girth 4 and is not K3,t . Let x, y be vertices of G such that the set of common neighbors, Z = {z1 , . . . , zk } has size at least 2. The existence of such a pair of points is guaranteed as the girth is 4 and there is an induced 4-cycle in G. It is worth mentioning that, since the girth of G is 4, none of the vertices adjacent to a vertex in Z are adjacent to either x or y. We now consider the component structure of G − {x, y}. Let Z1 , . . . , Zkz be the vertex sets for components that contain an element of Z and let X1 , . . . , Xkx , (respectively Y1 , . . . , Yky ) be the vertex sets for the components such that E(Xi , {x}) 6= ∅ and Xi ∩ Z = ∅ (respectively, E(Yi , {y}) 6= ∅ and Yi ∩ Z = ∅). We note that the collection of vertex  sets {X1 , . . . Xkx } and Y1 , . . . , Yky are not necessarily distinct, however, this potential S duplicate naming will not affect our subsequent analysis. We will show the desired cycle is in {x, y} ∪ i Zi . We note the components induced by any Xi or Yj do not provide an obstruction. This is easy to see as by definition any vertex in Xi or Yj is incident to precisely one of {x, y} as otherwise it would belong to Z. Thus G[Xi ] and G[Yj ] have minimum degree at least 2, and thus, contain a cycle. As a consequence, we may restrict our attention to the components G[Z1 ], . . . , G[Zkz ] without loss of generality. We first consider the case where one of the components, say T = G[Z1 ], is a tree. As the minimum degree of G is 3 and Z contains every vertex adjacent to both x and y, the leaves of T are given by Z1 ∩ Z. Let z, z 0 be two vertices of maximum distance in T and let the unique path between them be given by z = t0 , t1 , . . . , t` , t`+1 = z 0 . We first consider the case where ` ≥ 2 and so t1 and t` are distinct vertices. We note that E({x, y} , {t1 , t` }) = ∅ as both t1 and t` are incident to elements of Z. Thus t1 and t` have degree at least 3 in T and thus there are vertices t01 and t0` that are incident to t1 and t` , respectively. By the maximality of the distance between z and z 0 in T and the uniqueness of the shortest path in trees, t01 and t0` are also leaves and hence in Z. But then consider the components of G − C, where C is the cycle {y, z, t1 , t01 }. Note that {x, z 0 , t` , t0` } is a cycle and disjoint from {y, z, t1 , t01 } and thus is present in G − C. Furthermore, the components of T − {z, t1 , t01 } as well as the components G[Z2 ], . . . , G[Zkz ] are all incident to x and thus form a single component which contains the cycle {x, z 0 , t` , t0` }. Hence, C is the desired cycle. Thus we may now assume that any tree component among G[Z1 ], . . . , G[Zkz ] has diameter 2 and a unique vertex not in Z. Suppose G[Z1 ] is such a component and let v be the unique element of G[Z1 ] not in Z. Since v is adjacent to an element of Z, we have that v is not adjacent to {x, y}, has degree at least 3, and {x, y, v} ∪ (Z1 ∩ Z) induces copy of K3,t in G. As G is not equal to K3,t , this implies that one of Z2 , X1 , or Y1 exists and is not empty. Suppose first that X1 exists and let z, z 0 ∈ Z1 ∩ Z. Consider the components of G − {y, z, v, z 0 }. As every element of Z − {z, z 0 } is incident to x, G[X1 ] contains a cycle, and there is an edge between X1 and x, we have that every component formed contains a cycle. A similar argument holds if Y1 exists by exchanging the roles of x and y. Finally, assume that no components of the type Xi or Yj exist, but Z2 exists. Let z, z 0 ∈ Z ∩ Z1 and consider the components of G − {y, z, v, z 0 } . As all the components Z3 , . . . , Zkz as well as the vertices of Z1 − {z, z 0 } are connected to x, in order to show that the all the components of G − {y, z, v, z 0 } have a cycle it suffices to show that G[Z2 ∪ {x}] contains a cycle. To that end, if |Z2 ∩ Z| = 1, then every vertex in Z2 − Z is adjacent to at most one vertex in {x, y} ∪ Z and hence G[Z2 − Z] has minimum degree 2 and a cycle. Otherwise, |Z2 ∩ Z| ≥ 2 and G[Z2 ∪ {x}] contains a cycle by taking a path between distinct elements of Z in G[Z2 ] joined by the vertex x. At this point, we may S assume without loss of generality that G[Z0i ] is not a tree for0 all i. Suppose that theSinduced graph G[ i Zi ] has S at least two vertices of degree 1, z, z . Then as z and z are on no cycles in G[ i Zi ], the components of G[ i Zi − {z, z 0 }] all contain cycles. This gives that {x, z, y, z 0 } is the desired 3 (a) Wheel (b) K3,t with dashed optional edges Figure 1. Examples from the infinite family of girth 3 counterexamples to Lemma 1. S cycle C. Thus we may assume that there is at most 1 vertex of degree 1 in G[ i Zi ], which we call z, and S let z 0 ∈ Z − {z}. As the induced subgraph G[ i Zi − {z, z 0 }] has at most one vertex of degree 1 (potentially the unique common neighbor of z and z 0 ), all of its components contain a cycle and again {x, z, y, z 0 } is the desired cycle. We may now assume that for i, the induced subgraph G[Zi ] is not a tree and every element of Z has degree at least 2 in the relevant component. Suppose that kz ≥ 2 and let z ∈ Z1 ∩ Z, z 0 ∈ Z2 ∩ Z. Every vertex in Z1 −Z is adjacent to at most one of {x, y, z, z 0 } and so has degree at least 2 in G[Z1 −{z, z 0 }], while the elements of Z ∩ Z1 − {z, z 0 } are incident to neither of z or z 0 , and thus also have minimum degree 2. This implies that G[Zi − {z, z 0 }] has minimum degree 2 for all i and thus contains a cycle, and hence {x, z, y, z 0 } is the desired cycle. Finally, we may now assume that there is a single component Z1 of G − {x, y} that contains all elements of Z and further, that component is not a tree and every element of Z ∩ Z1 has degree at least two in the component. Now fix two elements z, z 0 ∈ Z and and let F be the forest of tree components of G[Z1 − {z, z 0 }]. Clearly if F is empty, then {x, z, y, z 0 } is the desired cycle. Thus we may assume that F is non-empty and let L be the leaves of F . As Z is an independent set and every vertex of z has degree at least 2 in G[Z1 ], we note that L ∩ Z = ∅. Further, as the minimum degree is at least 3, any ` ∈ L is adjacent to at least two of {x, y, z, z 0 }. As ` 6∈ Z, it can not be adjacent to both x and y. Additionally, ` can not be adjacent to one of {x, y} and one of {z, z 0 } as this forms a triangle. Thus every element of L is adjacent to both z and z’. Now suppose that there is some tree T ∈ F such that E(T, {x, y}) = ∅ and let `, `0 be leaves of that tree. But then, z and z 0 are antipodal points in a 4-cycle such that their common neighbors are in distinct components of G − {z, z 0 }. Specifically, `, `0 are in a different component than {x, y} and thus by previous arguments the desired cycle exists. Thus we may assume that every component of F is adjacent to either x or y. But then, if |Z| ≥ 4, we have that for any two leaves `, `0 ∈ L, the cycle {z, `, z 0 , `0 } is the desired cycle. Specifically, if z̄, z̄ 0 ∈ Z − {z, z 0 } then the tree components of G − {x, y, z, z 0 , `, `0 } are all connected to the cycle {x, z̄, y, z̄ 0 } via either x or y. To complete the proof we note that the common neighbors of z and z 0 include {x, y} and L, and thus have at least 4 elements. Thus, by repeating the arguments above with {z, z 0 } in the role of {x, y}, we may assume that |Z| ≥ 4 as needed.  One might hope Lemma 1 could be extended to girth 3 graphs with a similarly small set of exceptions as the girth 4 case. However, it is relatively easy to construct infinite families of counterexamples. For example, the wheel graph on (see Figure 1) on any number of vertices forms a counterexample as every 3-cycle in the graph involves the central hub as well as an edge from the from the outer cycle. Thus the removal of a 3-cycle destroys every cycle in the graph. Another infinite family of counterexamples can be formed by taking K3,t and adding a non-empty set of edges to the partition of size 3. In this case, every 3-cycle uses 2 of the vertices of partition of size 3 and thus because the graph is bipartite there are not enough vertices remaining on that side to form a cycle (see Figure 1). For d-regular graphs, the only counterexamples the authors are aware of are K4 and K5 . Thus it is possible there is a finite set of exceptions for a d-regular, girth 3, version of Lemma 1. 4 Theorem 1. Let G be a d-regular graph with d ≥ 5 and second largest adjacency eigenvalue λ. If the cyclic edge-connectivity exists, the girth is at least 4, and   2 2(d − 2)g ≤ n0 d − ,g d−λ r−1 where r = bg/2c, then the cyclic edge-connectivity of G is at least (d−2)g. If the girth is 3 and λ ≤ d−6+ 12/d, then the cyclic edge-connectivity of G is at least 3d − 6. Proof. Let S be a minimal set of edges such that G − S is disconnected with all components containing a cycle and that |S| < (d − 2)g. By minimality, we may assume that there is some set of vertices X such that S = E(X, X), |X| ≤ X , and H = G[X] contains a cycle. Further, we may assume that X is the minimal cardinality set yielding an edge cut of size |S|. Now, by the Discrepancy Inequality, we have that |X| X |X| ≥ (d − λ) . n 2 Thus, to prove the desired result for g ≥ 4 it suffices to show that   2(d − 2)g 2 ,g ≥ . |X| ≥ n0 d − r−1 d−λ e(X, X) ≥ (d − λ) We first observe that if H is a simple cycle, then |X| ≥ g and e(X, X) ≥ (d − 2)g, as desired. Thus assume without loss of generality that H is not a cycle. Furthernote  that, if x ∈ X is such that there is some cycle C ∈ G[X − {x}], then the degree of x in H is at least d2 as otherwise G[X − {x}] contains the cycle C, G[X ∪ {x}] contains the same cycle as G[X], and e(X − {x} , X ∪ {x}) ≤ e(X, X). As a consequence, the minimum degree in H is at least 2. At this point it is possible to observe that H is 2-edge-connected. Specifically, suppose that there exists two disjoint sets X1 and X2 such that X1 ∪ X2 = X and e(X1 , X2 ) ≤ 1. As the minimum degree in H is at least 2, both G[X1 ] and G[X2 ] contain some cycle. By minimality of the edge cut and that X1 , X2 ⊂ X, we have that e(X1 , X1 ), e(X2 , X2 ) ≥ e(X, X) + 1, thus e(X, X) = e(X1 , X1 ) + e(X2 , X2 ) − 2 ≥ 2e(X, X), a contradiction. Since H is 2-edge-connected, there exists an ear decomposition for H [12]. Specifically, there a Sexists  i−1 cycle C ∈ H as well as paths P1 , . . . , Pk such that the internal vertices of Pi are disjoint from C ∪ j=1 Pj S  k and H = C ∪ j=1 Pj . Since we may assume that H is not a cycle, we have that there is a non-zero number of paths in the ear decomposition. So we may consider the last path in the ear decomposition, Pk . By construction of the ear decomposition, any internal vertex in Pk will  have degree 2 in H but will be disjoint from the cycle C. But as such vertices have degree at least d2 and d ≥ 5, no such vertex exists and Pk is a single edge e = {x, y}. By the properties of the ear decomposition H − e is a 2-edge-connected graph and hence there are at least two, edge-disjoint, paths between x and y in H − e, denote them by Q and Q0 . Without loss of generality we assume that the total length of Q and Q0 is minimized. Now suppose there exists vertices a and b that are on both Q and Q0 , but in opposite orders. That is, Q = xQ1 aQ2 bQ3 y and Q0 = xQ01 bQ02 aQ03 y. We can then construct two new walks from x to y, xQ1 aQ03 y and xQ01 bQ3 y which have total shorter length. Thus, if x = a0 , a1 , . . . , at−1 , at = y are the intersection points of Q and Q0 , they occur in the same order in both Q and Q0 . As a consequence, Q ∪ Q0 can be thought of as a series of vertex incident cycles C0 , C1 , . . . , Ct such that aj , aj+1 ∈ Cj and Ci−1 ∩ Ci = {ai }. Now for every vertex v in H (except potentially   x, y if t = 1 and a1 if t = 2), there is some cycle not containing v and hence the degree of v is at least d2 ≥ 3. As the degree of x and y are at least 2 and the degree of a1 is at least 4, this implies that average degree of H is at least 2 +  for some  > 0. As H has average degree 2 +  and girth at least g, by the Irregular Moore Bound, we have that |X| ≥ n0 (2 + , g). But then, since G is d-regular and e(X, X) < (d − 2)g, we have the average degree is at least d− (d − 2)g . n0 (2 + , g) 5 In particular, we have that  must satisfy that (d − 2 − )n0 (2 + , g) < (d − 2)g. Consider first the case where g = 2r ≥ 4, and note that   r−1 X (d − 2 − )n0 (2 + , 2r) = (d − 2 − ) 2 (1 + )j  j=0 (1 + )r − 1     r ≥ 2(d − 2 − ) r +  . 2 = 2(d − 2 − ) Thus we need to have     r 2(d − 2 − ) r +  < 2r(d − 2), 2 which can be rearranged to     r 2 −  < 0.  d−2− r−1 2  2 As we already have that  > 0, this implies that  > d − 2 − r−1 and thus |X| ≥ n0 d − Finally consider the case where g = 2r + 1 ≥ 5. In this case we have that   r−1 X (d − 2 − )n0 (2 + , 2r + 1) = (d − 2 − ) 1 + (2 + ) (1 + )j  2 r−1 , 2r j=0  ≥ (d − 2 − ) 1 + (2 + ) r−1 X  (1 + j) j=0      r = (d − 2 − ) 1 + (2 + ) r +  2         r r 2 = (d − 2 − ) 1 + 2r + 2 +r +  2 2     r 2 = (d − 2 − ) 1 + 2r + r2  +  2 Thus, we have that  satisfies that        
r r 2  (d − 2)r2 − g + (d − 2) − r2  −  < 0. 2 2 Letting       r 2 r 2 f () = −  + (d − 2) − r  + (d − 2)r2 − g, 2 2 it is easy to see that lim f () = −∞, →−∞ f (0) = (d − 2)r2 − 2r − 1 > 0, f (d − 2) = −g, and lim f () = −∞ →∞ 6  . Figure 2. The graph from Example 1. The edge cut of size 8 can clearly be seen. Thus f () has one root in (−∞, 0) and one in (0, d − 2). Let ∗ be the root of f () in (0, d − 2), then we have that  ≥ ∗ and |X| ≥ n0 (2 + ∗ , 2r + 1). Observing that,       2    r 2 r 2 2 + (d − 2) − r2 f d−2− =− d−2− d−2− + (d − 2)r2 − g 2 r−1 2 r−1 r−1 !    2    r 2 2 2r 2(d − 2)r g = − d−2− + d−2− d−2− + − r
2 r−1 r−1 r−1 r−1 2 !     r 2 2 − 2r 2(d − 2)r g = d−2− + − r
2 r−1 r−1 r−1 2    r 2(d − 2) 4r − 4 4r + 2 + − = 2 r−1 (r − 1)2 r(r − 1) 2(d − 2)(r − 1)r + (4r − 4)r − (4r + 2)(r − 1) = 2(r − 1) 2(d − 2)(r − 1)r + 2 − 2r = 2(r − 1) = (d − 2)r − 1 >0 Thus ∗ > d − 2 − 2 r−1  and |X| > n0 d − 2 r−1 , 2r  +1 . Finally, the case for g = 3 proceeds similarly as above except that n0 (2 + , 3) = 3 +  and thus ∗ can be determined to be exactly d − 5.  Combining these results yields the following spectral condition for when the cyclic edge-connectivity of a graph is given by a cut induced by a minimum length cycle. Corollary 1. If G is a d-regular graph with d ≥ 5, girth g ≥ 4, and the second largest eigenvalue of the adjacency matrix, λ, satisfies   2(d − 2)g 2 ≤ n0 d − ,g , d−λ r−1 where r = bg/2c, then the cyclic edge-connectivity is (d − 2)g. Given the relatively weak spectral condition required in Theorem 1 and the extensive use of the Irregular Moore Bound in the proof, one might naturally wonder whether any spectral condition is required at all. Here we briefly provide an example that shows that, for girth 4 at least, the spectral condition is required. Example 1. We will construct a 5-regular graph on 48 vertices which has cyclic edge-connectivity at most 8 < 3 × 4 and whose second largest eigenvalue is at least 4.5 > 5 − n0 12 (3,4) . Let G be the graph obtained by the following: first construct two hypercubes on 16 vertices, the first on a0 , a1 , ..., a15 and the second on d1 , d2 , ..., d15 , numbered according to hypergraph structure. Then add 8 vertices b0 , ..., b7 and add an edge from bi to ai for each i and an edge from bi to ai+9 for i = 0, ..., 6 and from b7 to a8 . Similarly, add 8 vertices c0 , ..., c7 and edges from ci to di for each i, from ci to di+9 for i = 0, ..., 7 and from c7 to d8 . Now add an 8-cycle in the obvious way among the vertices b0 , ..., b7 and another among 7 the vertices c0 , ..., c7 . Finally, add an edge between bi and ci for each i. This graph has girth 4, since the hypercube had girth 4, and the way we added edges introduced no 3-cycles. As remarked above, there is an edge cut of size 8 (the edges bi ci for i = 0, ..., 7) that separates the graph into two components, each of which has a cycle. Direct computation of the eigenvalues of the adjacency matrix shows that the second largest eigenvalue is approximately 4.56. This graph is pictured in Figure 2. We remark that the above example can be expanded as long as we can connect the added cycles to the hypercubes in a regular way that does not produce any triangles. For instance, this can be done with 11-cycles instead of 8-cycles to produce a graph with cut size 11. Interestingly, this example has the same second largest eigenvalue as the graph in the previous example. We omit the details. Acknowledgements. The authors would like to thank Carlos Ortiz-Marrero for helpful discussions. References [1] A. Abiad, B. Brimkov, X. Martinez-Rivera, S. O, and J. Zhang, Spectral bounds for the connectivity of regular graphs with given order, Electronic Journal of Linear Algebra, 34 (2018), pp. 428–443. [2] N. Alon and F. Chung, Explicit construction of linear sized tolerant networks, Discrete Mathematics, 72 (1988), pp. 15–19. [3] N. Alon, S. Hoory, and N. Linial, The moore bound for irregular graphs, Graphs and Combinatorics, 18 (2002), pp. 53– 57. [4] N. Biggs, Algebraic graph theory, Cambridge Mathematical Library, Cambridge University Press, Cambridge, second ed., 1993. [5] F. T. Boesch, Large-scale networks, theory and design, IEEE press, 1976. [6] B. Bollobás, Extremal graph theory, Courier Corporation, 2004. [7] A. Bondy, Graph Theory: (Graduate Texts in Mathematics), Springer, 2010. [8] G. A. Dirac, Some results concerning the structure of graphs, Canad. Math. Bull., 6 (1963), pp. 183–210. [9] F. Harary, Conditional connectivity, Networks, 13 (1983), pp. 347–357. [10] R. J. Lipton and R. E. Tarjan, A separator theorem for planar graphs, SIAM Journal on Applied Mathematics, 36 (1979), pp. 177–189. [11] L. Lovász, On graphs not containing independent circuits, Mat. Lapok, 16 (1965), pp. 289–299. [12] H. E. Robbins, A theorem on graphs, with an application to a problem of traffic control, The American Mathematical Monthly, 46 (1939), pp. 281–283. Pacific Northwest National Laboratory, Richland, WA, 99354 E-mail address: [email protected] Mathematics Department, Brigham Young University, Provo, UT, 84602 E-mail address: [email protected] Pacific Northwest National Laboratory, Richland, WA, 99354 E-mail address: [email protected] 8