Regular Graphs with Minimum Spectral Gap
M. Abdi, E. Ghorbani, W. Imrich

TL;DR
This paper proves Aldous and Fill's conjecture on the spectral gap for cubic graphs and characterizes the structure of quartic graphs with minimal spectral gap, advancing understanding of spectral properties in regular graphs.
Contribution
It confirms the spectral gap conjecture for cubic graphs and describes the structure of quartic graphs with minimal spectral gap.
Findings
Spectral gap bound is proven for cubic graphs.
Quartic graphs with minimal spectral gap have a path-like structure.
The conjecture relates relaxation time to spectral gap in regular graphs.
Abstract
Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with vertices is . This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected -regular graph on vertices is at least , and the bound is attained for at least one value of . Based upon previous work of Brand, Guiduli, and Imrich, we prove this conjecture for cubic graphs. We also investigate the structure of quartic (i.e. 4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks.
| situation of | situation of next few vertices after appropriate switchings |
|---|---|
| (i) | |
| (ii) | or returning to (i) |
| (iii) | or turns to a cut vertex |
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Taxonomy
TopicsGraph theory and applications · Markov Chains and Monte Carlo Methods · Nanocluster Synthesis and Applications
Regular Graphs with Minimum Spectral Gap
M. Abdi E. Ghorbani W. Imrich
aDepartment of Mathematics, K. N. Toosi University of Technology,
P. O. Box 16765-3381, Tehran, Iran
bSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P. O. Box 19395-5746, Tehran, Iran
cMontanuniversität Leoben, Leoben, Austria
Abstract
Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with vertices is . This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected -regular graph on vertices is at least , and the bound is attained for at least one value of . Based upon previous work of Brand, Guiduli, and Imrich, we prove this conjecture for cubic graphs. We also investigate the structure of quartic (i.e. 4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks.
Keywords: Spectral gap, Algebraic connectivity, Relaxation time, Cubic graph, Quartic graph
AMS Mathematics Subject Classification (2010): 05C50, 60G50
††footnotetext: E-mail Addresses: [email protected] (M. Abdi), [email protected] (E. Ghorbani), [email protected] (W. Imrich)
1 Introduction
All graphs we consider are simple, that is undirected graphs without loops or multiple edges. The difference between the two largest eigenvalues of the adjacency matrix of a graph is called the spectral gap of . If is a regular graph, then its spectral gap is equal to the second smallest eigenvalue of its Laplacian matrix and known as algebraic connectivity.
In , Bussemaker, Čobeljić, Cvetković, and Seidel ([4], see also [5]), by means of a computer search, found all non-isomorphic connected cubic graphs with vertices. They observed that when the algebraic connectivity is small the graph is long. Indeed, as the algebraic connectivity decreases, both connectivity and girth decrease and diameter increases. Based on these results, L. Babai (see [9]) made a conjecture that described the structure of the connected cubic graph with minimum algebraic connectivity. Guiduli [9] (see also [8]) proved that the cubic graph with minimum algebraic connectivity must look like a path, built from specific blocks. The result of Guiduli was improved as follows confirming the Babai’s conjecture.
Theorem 1.1** (Brand, Guiduli, and Imrich [3]).**
Among all connected cubic graphs on vertices, , the graph (given in Figure 1) is the unique graph with minimum algebraic connectivity.
The relaxation time of the random walk on a graph is defined by , where is the second largest eigenvalue of the transition matrix of , that is the matrix in which and are the diagonal matrix of vertex degrees and the adjacency matrix of , respectively. A central problem in the study of random walks is to determine the mixing time, a measure of how fast the random walk converges to the stationary distribution. As seen throughout the literature [2, 6], the relaxation time is the primary term controlling mixing time. Therefore, relaxation time is directly associated with the rate of convergence of the random walk.
Our main motivation in this work is the following conjecture on the maximum relaxation time of the random walk in regular graphs.
Conjecture 1.2** (Aldous and Fill [2, p. 217]).**
Over all connected regular graphs on vertices, .
In terms of the eigenvalues of the normalized Laplacian matrix, that is the matrix , the Aldous–Fill conjecture says that the minimum second smallest eigenvalue of the normalized Laplacian matrices of all connected regular graphs on vertices is . This can be rephrased in terms of the spectral gap as follows, giving another equivalent statement of the Aldous–Fill conjecture.
Conjecture 1.3**.**
The spectral gap (algebraic connectivity) of a connected -regular graph on vertices is at least , and the bound is attained at least for one value of .
It is worth mentioning that in [1], it is proved that the maximum relaxation time for the random walk on a connected graph on vertices is settling another conjecture by Aldous and Fill ([2, p. 216]).
In [3], it is mentioned without proof that the algebraic connectivity of the graphs (of Theorem 1.1) is , where its proof is postponed to another paper which has not appeared. We prove this equality, thus, showing that the minimum spectral gap of connected cubic graphs on vertices is , which implies the Aldous–Fill conjecture for . As the next case of the Aldous–Fill conjecture and as a continuation of Babai’s conjecture, we investigate the connected quartic, i.e. -regular, graphs with minimum spectral gap. We show that similar to the cubic case, these graphs must have a path-like structure with specified blocks (see Theorem 3.1 below). Finally, we put forward a conjecture about the unique structure of the connected quartic graph of any order with minimum spectral gap.
2 Minimum spectral gap of cubic graphs
In this section, we prove that the minimum spectral gap of connected cubic graphs on vertices is .
Let be a graph on vertices and be its Laplacian matrix. For any , the value is called a Rayleigh quotient. We denote the second smallest eigenvalue of known as the algebraic connectivity of by . It is well known that
[TABLE]
where is the all- vector. An eigenvector corresponding to is known as a Fiedler vector of . In passing we note that if , then
[TABLE]
where is the edge set of .
Considering the graphs of Theorem 1.1, we let (numbered consecutively from left to right) be a partition of the vertex set such that each cell has size 1 or 2, consisting of the vertices drawn vertically above each other as depicted in Figure 1. We note in passing that partition is a so-called ‘equitable partition’ of .
Lemma 2.1** ([3]).**
Let be a Fiedler vector of .
- (i)
Then the components of on each cell of the partition are equal.
- (ii)
Let be the values of on the cells of . Then the form a strictly monotone sequence changing sign once.
Recall that a block of a graph is a maximal connected subgraph with no cut vertex—a subgraph with as many edges as possible and no cut vertex. So a block is either (a trivial block) or is a graph which contains a cycle. If a graph has no cut vertex, then itself is also called a block. The blocks of a connected graph fit together in a tree-like structure, called the block tree of . The block tree of the graphs are paths which justifies the description ‘path-like structure.’
We now present the main result of this section.
Theorem 2.2**.**
The minimum algebraic connectivity of cubic graphs on vertices is .
Proof.
In view of Theorem 1.1, it suffices to show that . To prove this, we consider two cases based on the value of mod .
Case 1.
In this case is the upper graph of Figure 1. Let be the number of non-trivial blocks of . So we have .
We first prove that is an upper bound for .
We define the vector with
[TABLE]
Note that is skew symmetric vector, i.e. , for , and so . We extend to define the vector on as shown in Figure 2.
The vector (like ) is a skew symmetric. It follows that . Therefore, by (1) we have
[TABLE]
Note that (2) is obtained using the identities and . For (3) we use the identities
[TABLE]
which are a consequence of the fact that .
We now prove that is a lower bound for .
Let be a Fiedler vector of . Let be the non-trivial blocks of , and be the set of edges of and be the set of all bridges of . Then we have
[TABLE]
The graph has cut vertices. Consider the components of on the cut vertices of together with the four components ; we define as the vector consisting of these components, as depicted in Figure 3.
Note that is skew symmetric. To verify this, observe that by the symmetry of , is also an eigenvector for . It follows that itself is a skew symmetric eigenvector for (note that from Lemma 2.1, it is seen that ), so that we may replace for . Now, from Lemma 2.1, it follows that . As is skew symmetric, it follows that is also skew symmetric and thus . Let be one of the middle blocks of , i.e. . The components of on the left vertex and the right vertex of are and , respectively. Let be the component of on the two middle vertices of (which are equal by Lemma 2.1) as shown in Figure 4.
Then
[TABLE]
The right hand side, considered as a function of , is minimized at . This implies that
[TABLE]
It follows that
[TABLE]
which in turn implies that
[TABLE]
We also have
[TABLE]
which holds because (cf. Lemma 2.1). Now, from (2), (5) and (6) we infer that
[TABLE]
Note that the right hand side of (7) is the Rayleigh quotient of for the path . Thus, by the fact that (see [7]), it follows that
[TABLE]
Therefore,
[TABLE]
Case 2.
In this case, is the bottom graph of Figure 1. We define the graph as shown in Figure 5. The symmetries of are similar to those of the graph . So the arguments of the previous case also work for , in particular has a skew symmetric Fiedler vector. Therefore, we have . Let be the Fiedler vector of with . We define the vector of length by
[TABLE]
where . It is seen that is orthogonal to . We label the vertices of by the components of as shown in Figure 5. We observe that . On the other hand,
[TABLE]
So , which means that the Rayleigh quotient for on is smaller than . It follows that . By a similar argument, we see that . Therefore, . ∎
3 Structure of quartic graphs with minimum spectral gap
Motivated by the Aldous–Fill Conjecture and also as an analogue to Babai’s conjecture, we consider the problem of determining the structure of connected quartic graphs with minimum spectral gap. We prove that such graphs have a path-like structure (see Figure 6) and specify their blocks. Finally, we pose a conjecture which precisely describes the connected quartic graphs with minimum spectral gap.
We remark that in a quartic graph, any cut vertex belongs to exactly two blocks and further has degree in each of them. Therefore, in the quartic graphs having a path-like structure, the middle and end blocks have exactly two and one vertices of degree , respectively.
One of our goals in this section is to specify the structure of the blocks of a quartic graph with minimum spectral gap. As we shall prove, the blocks of such graphs are of two types: ‘short’ and ‘long’. By short blocks we mean those given in Figure 7.
The long blocks, roughly speaking, are constructed by putting some short blocks together with the general structure given in Figure 8.
More precisely, the building ‘bricks’ of long blocks are the graphs , obtained by removing the right degree vertex of the corresponding short blocks, and the graphs , obtained by removing both degree vertices of . For any of these graphs, say , we denote its mirror image by . A long block is constructed from some bricks , where each is joined by two edges to (as shown in Figure 8). There are three types of long blocks:
- (i)
long end block: , , and ;
- (ii)
long middle block: , , and ;
- (iii)
long complete block: , , and
We note that long complete blocks are quartic and long end blocks and middle blocks have exactly one or two vertices of degree , respectively.
Here is the main result of this section.
Theorem 3.1**.**
Let be a graph with the minimum spectral gap in the family of connected quartic graphs on vertices. If is a block, then either and is one of the graphs of Figure 9, or and is a long complete block. If itself is not a block, then it has a path-like structure in which each left end block is either one of or a long end block, and each middle block is either one of or a long middle block. Each right end block is the mirror image of some left end block.
Subsection 3.2 is devoted to the proof of Theorem 3.1. In fact, Theorem 3.1 follows from Theorems 3.11 and 3.15 below.
3.1 Elementary moves and their effect on algebraic connectivity
In this subsection we present the main tool of the proof of Theorem 3.1, that is, a local operation on edges of a graph which preserves the degree sequence of the graph.
Let be a graph. By ‘’ and ‘’ we denote, respectively, adjacency and non-adjacency in . An elementary move or switching in is a switching of parallel edges: let and , then the elementary move denoted by removes the edges and and replaces them by the edges and .
Definition 3.2**.**
Let be a graph and be a Fiedler vector of , considered as a weighting on the vertices; for we write . For convenience, we may assume the vertex set is and that the vertices are numbered so that . We call this a proper labeling of the vertices (with respect to the eigenvector ).
The following two lemmas were initially used by Guiduli [9] (see also [8]) for cubic graphs but they also hold for quartic graphs.
Lemma 3.3**.**
Let be a connected graph. Let be a Fiedler vector of . If there are vertices in such that , , , , with , and , then does not increase the algebraic connectivity.
Definition 3.4**.**
A switch or elementary move is said to be proper if it satisfies the conditions of Lemma 3.3. In particular, with proper labeling on the vertices, is proper if and .
We will use proper switchings to transfer the graphs into the path-like structure without increasing the algebraic connectivity. The following lemma keeps the graph connected during this procedure.
Lemma 3.5**.**
Let be a properly labeled connected graph on and . Assume that is disconnected and that each of its components has an edge which is not a bridge. Then we may reconnect the graph using proper elementary moves to make connected, not increasing the algebraic connectivity.
In the arguments which follow, we use proper elementary moves to connect two specific vertices and . The following remark demonstrates when such a switch does, or does not, exist.
Remark 3.6**.**
Let be a graph whose vertices are properly labeled and be two vertices of with . Suppose we are looking for a proper switch to connect and without altering the induced subgraph on . From Lemma 3.3 it is evident that such a switch does not exist if and only if any neighbor of in is adjacent to any neighbor of in .
3.2 Proof of Theorem 3.1
Theorem 3.1 follows from Theorems 3.11 and 3.15, which will be proved in this subsection.
Hereafter, we assume that is a connected quartic graph with vertices, whose vertices are labeled properly as described in Definition 3.2. Our goal is to utilize proper elementary moves to transfer to one of the graphs described in Theorem 3.1.
3.2.1 The subgraph on the first few vertices
Our first goal is to prove that we can reconnect (by proper elementary moves) the first few vertices of to get one of the four subgraphs , , , .
In the process of reconnecting , we are usually in the situation that for some , we have already built some specific subgraph on and continue to build a desired subgraph on in a way not to alter the subgraph already constructed on . The next two lemmas deal with such situations.
Lemma 3.7**.**
Let be a connected graph with vertices where all the vertices have degree except the first two, which have degree . Then, by proper switchings, can be transferred into a graph in which , or and can be transferred into or , respectively.
Proof.
If some neighbor of is not adjacent to some neighbor of , then connects to . Otherwise any neighbor of is adjacent with any neighbor of . This is only possible in two cases: (i) and share three neighbors all of which are adjacent to each other or (ii) is the graph of Figure 10a. If (i) is the case, then and is . In the case (ii), with no loss of generality, we assume that , , and . We first and then , which result in Figure 10b. Now we perform , and then either if or if . As a result we obtain (the outcome in the case is shown in Figure 10c). ∎
Lemma 3.8**.**
Let be a graph with vertices where all the vertices have degree except for which have exactly two neighbors among . Further . Then by proper switchings, can be transferred into a graph with , or and can be transferred into the the graph (depicted in Figure 11b).
Proof.
First suppose that . If , then has a neighbor which is not adjacent with (otherwise should have degree ). Then makes and adjacent.
Next suppose that . If some neighbor of is not adjacent to some neighbor of , then connects to . Otherwise and must share three neighbors all of which are adjacent to each other. In this case, the neighbors of and the neighbors of cannot be adjacent. So there is a switch to connect to and we are done. Therefore we suppose that . If some neighbor of is not adjacent with some neighbor of , then we are done. Otherwise either we are in the situation of Figure 11a and so by , , which implies as discussed above; or and share two neighbors, say , such that and is adjacent to both and , which means that and is the graph given in Figure 11b. Note that and can be either adjacent or non-adjacent, which is illustrated by dashed edges in Figure 11. ∎
Now, we can prove that the first few vertices of can be reconnected to induce the subgraphs asserted in Theorem 3.1.
Lemma 3.9**.**
The induced subgraphs on the first few vertices in can be transferred by proper switchings into one of the four subgraphs , , , . Furthermore, if , then can be transferred into one of the graphs , or .
Proof.
In Steps 1–5 below, we show that the induced subgraph on the first five to seven vertices of can be transferred into or to one of the subgraphs given in Figure 12, or has at most 9 vertices and it is one of the graphs , or . In the final Step 6, from we obtain one of .
Step 1. Connecting to and to . Assume that is not adjacent with some . So has some neighbor . It is not possible that is adjacent with any neighbor of (this requires having degree ). So there is some vertex such that and . Now, makes 1 adjacent to .
We may assume that is connected. The desired switch to is possible, unless and share three neighbors in and all the three neighbors are adjacent to each other. But this contradicts the fact that is connected.
Step 2. Connecting to . Again we may assume by Lemma 3.5 that is connected. If no proper switch leading to exists, then, similarly to Step 1, we see that . Also, we may assume that , because otherwise has a neighbor with , and so connects to . Let be a neighbor of other than and . First suppose that . If , then by Remark 3.6 the desired switch exists, except in the two situations (a) and (b) of Figure 13. For (a), we are done by . Note that (b) is not possible in view of the fact that is connected. If , then the desired switch exists, except in the situation (c) of Figure 13 for which implies . So we are left with the case that . If , then . Otherwise, are all the neighbors of . Let be the fourth neighbor of . Then .
Step 3. Connecting to . Let be the fourth neighbor of . We consider the following two cases.
- (i)
. Let and be the other two neighbors of . If or , then the desired switch is possible. Otherwise we are in the situation of Figure 14a. We first show that or . If , then by examining the neighbors of and , proper switches to or will clearly exist. If , then the desired switch will exist, except when , , and in which case and . So we have that or and thus we are in either of the situations (b) or (c) of Figure 14. (Note that if there is no edge in (b) or in (c), then it is easy to find a switch that connects to .) For (b), has a neighbor and . Then . For (c), has a neighbor and . Then . It turns out that both (b) and (c) lead to the subgraph (d) of Figure 14. If both and are adjacent to , then and we get . Therefore we suppose that both and cannot be adjacent to . Then either or , for which we apply or , respectively.
- (ii)
. If the remaining neighbors of and are not the same, then the desired switch is available. Otherwise, similarly to (i), we have or . So in view of Remark 3.6, we are in either of the situations (a) or (b) of Figure 15. For (a), first let . If , then , and if , then connects to , which reduces the graph to Case (i). Now let . If and , then , and by and then we transfer to . If or , then there is a neighbor of such that either and , and then , or and . Then connects to . We do the same for (b) to connect to . So both (a) and (b) lead to the subgraph (c) of Figure 15. Now connects to , which reduces the graph to Case (i).
Step 4. Connecting to . Let be a neighbor of . If , we may choose so that , and then . So assume that . From Remark 3.6, it is seen that the desired switch is available, except in the situations of Figure 16. For each of them, we first show that . Then, with this edge, the desired switches can be found. In the one in Figure 16a, if is adjacent to both and , then and . Otherwise has a neighbor and . We first and then . The other two situations are similar. Note that in Figure 16b, if and , then and ; and in Figure 16c, if , , and , then . By and then , we can thus transfer to .
Step 5. So far we obtain on the first five vertices either the subgraph (a) or (b) of Figure 17. If (a) is the case, letting to be the fourth neighbor of , then connects to , and so we obtain . Now, assume that (b) is the case. If we can find a switch to connect to , we again reach . Otherwise, it is easily seen that by proper switching we can connect to as shown in Figure 17c. Furthermore, if we can find a suitable switch to connect to , we reach the graph of Figure 12. Otherwise, it is easily seen by switching that and that we can reach Figure 17d. If there is no switch to connect to , then we can find a proper switch to connect to , except when all the three vertices , , and are adjacent to and and , in which case and . Now, from (d), if , then connects to and thus is obtained. Otherwise and we reach the graph of Figure 12.
Step 6. So far we have obtained one of the subgraphs , or of Figure 12, unless , in which case we obtained the graphs of Figure 9. We show that continuous reconnecting, starting from and , leads to , , , or .
First, consider . We have either or . Let . It is easy to find a switch that connects to . If further , then we have the block . If , it is easily seen, by switching, that . Then reduces the subgraph on to . Now, let . By switching it is seen that and . Thus we are in the situation of Lemma 3.8 for , which leads to either of the subgraphs (a) or (b) of Figure 18. Now, reduces (b) to the subgraph of Figure 18c, which is the unique graph of Theorem 3.1 on 10 vertices. For (a), first let . If further , then we get , otherwise and then reduce the subgraph on to . Now, let . Then reduces the subgraph on to .
Secondly, consider . We have either or . First let . It is easy to find a switch that connects to . If further , then we obtain the block . If , then reduces the graph to . Now, let . Then reduces the subgraph on to .
∎
3.2.2 General Steps
In this section, we continue reconnecting by proper switchings to construct the middle and end blocks with the structure described in Theorem 3.1.
Lemma 3.10**.**
Let be a graph with vertices where all the vertices have degree except for , which has degree . If , then, by proper switchings, we can transform the induced subgraph on the first four or five vertices into one of the subgraphs given in Figure 19. If , then can be transformed into either of or .
Proof.
First we show that . To see this, assume that is not adjacent with some . So has some neighbor . It is not possible that is adjacent with any neighbor of (this requires having degree ). So there is some vertex such that and . Now, makes adjacent to . The graph satisfies the conditions of Lemma 3.7 with , and so if , can be transferred into which means that can be transferred into , respectively. For , nothing remains to be proved and so we assume that . Hence, from Lemma 3.7, it follows that . By the same argument given above for , we see that is adjacent with both and . Now satisfies the conditions of Lemma 3.8 and so if (i.e. ), can be transferred into (with ) and so to . Therefore, we assume that . Thus from Lemma 3.8 it follows that . So far we have obtained the desired subgraph on , which is the same on all the graphs of Figure 19.
To conclude the proof, we consider the possible adjacencies between the three vertices , and . If , we are done as we obtain either the subgraph (a) or (b) of Figure 19. So, in what follows we assume that . We claim that . By Lemma 3.5, we may assume that is connected. Let be the fourth neighbor of . If , the desired switch is available, except in the case (a) of Figure 20 (in which case and the block is obtained). If , by Remark 3.6 the desired switch is available in any situation other than the case (b) of Figure 20. Then either or for which by either or , respectively, we have and the claim follows. Again we may assume that is connected. Our goal is to show that and and thus we will come up with either of the subgraphs (c) or (d) of Figure 19. As above is the fourth neighbor of . There are two possibilities:
- (i)
. A switch to connect to exists, except in the situation of Figure 20c. If , then we are done by reaching the subgraph given in Figure 19c. If , let and be the other neighbors of . Then and give rise to the subgraph of Figure 19c again. If and , then has two neighbors and that are non-adjacent to . Then by and the subgraph of Figure 19d is obtained.
- (ii)
. A switch to connect to exists, except in the situation of Figure 20d. If , then we are done by reaching the subgraph of Figure 19d. Otherwise, in a similar manner, the switches which give rise to the subgraph of Figure 19d can be found easily by examining the adjacencies between the neighbors of and (or ).
∎
We are now in a position to prove the ‘first half’ of Theorem 3.1, that is to show that can be transferred to one of the graphs of Theorem 3.1.
Theorem 3.11**.**
By proper switchings, any connected quartic graph can be turned into one of the graphs described in Theorem 3.1.
Proof.
For the assertion is proved in Lemma 3.9. So we may assume that . We start rebuilding on its first few vertices as in Lemma 3.9. As we saw there, the first few vertices of can be transformed into one of the subgraphs . Moreover, whatever we obtained, we ended up either with a cut vertex, or with one of the situations (i) or (ii) of Table 1. Since cut vertices in a quartic graph have necessarily degree , we can employ Lemma 3.10 for reconnecting following a cut vertex. Doing so, we again reach at one of the situations (i), (ii), or (iii) of Table 1. We now demonstrate what can be constructed afterwards. As verified below, by proper switchings, the situation of the next few vertices can be determined from the situation of according to Table 1.
In Case (i) it is easily seen, by switching, that . If further , then we obtain the first possible outcome. If , it is easily seen, by switching, that . Now, we can employ Lemma 3.7 (with ), which implies that either or either of or as an end block can be obtained.
In (ii), we assume that , otherwise we return to Case (i). It is easily seen, by switching, that and . Then, satisfies Lemma 3.8 and so that either , or we obtain the third possible outcome, in which case we either get or we are left with one of the two situations (a) or (b) of Figure 21. For (a), by and then , and for (b), by , we obtain . So we assume that . If we further have , we come up with the first possible outcome. So assume that . Then it easy to find a switch that ensures . If and , then we obtain the second possible outcome. Otherwise, we have either or , and then or , respectively, ensures that , which return us to Case (i).
In (iii), it is easily seen, by switching, that and , as shown in Figure 21c. If , then, by , is turned to a cut vertex . Now, let . If further , then we obtain the first outcome, otherwise by , is turned into a cut vertex .
The outcome of Table 1 is either an end block or, after proper reconnecting, we are again in one of the situations (i), (ii), (iii). Therefore, we may keep repeating this until we end up with an end block.
We need further switchings to transform the blocks into the structure asserted in Theorem 3.1. Two types of structures may still appear in our graph: X-shape (Figure 22a) and X*′-shape (Figure 22b). The X′*-shape, in which and , should be avoided. We can simply remove it by , which transfers it to Figure 22c. For X-shapes the situation is different. They should only appear in specific places, namely in the short blocks or in an or as the first brick or the last brick in a long block, respectively.
First note that if we have two consecutive X-shapes as in Figure 23a, then by , we can transfer it to Figure 23d. If in an X-shape, the two right vertices are adjacent and it is neither in an -block, nor in an (as the last brick in a long block), then it must be in the situation of Figure 23b, which can be transferred by to Figure 23e. If in an X-shape, the two left vertices are adjacent and it is neither in an -block, nor in an (as the first brick in a long block), then it must be in the situation of Figure 23c, which can be transferred by to Figure 23f.
The above arguments show that can be transformed into one of the graphs described in Theorem 3.1. ∎
3.2.3 Final Step
Let denote the family of graphs described in Theorem 3.1. To complete the proof of Theorem 3.1, we need to show that all connected quartic graphs with minimum algebraic connectivity belong to . In fact, it might be possible that is transformed (by means of proper switchings) to a graph , where we still have . We show that, under these circumstances, must be isomorphic to .
Remark 3.12**.**
Considering the structure of the graphs , we regard the vertices drawn vertically above each other as a cell. The cells of , in fact constitute an ‘equitable partition’ of . Each cell contains one or two vertices (except for the first cells in , or some cells in the ’s (of Figure 9) that have size and , respectively). Further, we know that the weights on the vertices of given by a Fiedler vector of are non-increasing from left to right. We may assume that the vertices that are in the same cell have the same weight. Otherwise, let be a vector obtained from by interchanging the weights of the vertices within all cells (in fact this is carried out by the action of an automorphism of , which also works for the first cells in ). Then and thus is an eigenvector corresponding to where is constant on each cell. Thus we may assume that is a non-increasing eigenvector for and is constant on each cell. The above argument may not work for , but for this small graph this can be done by direct inspection.
Lemma 3.13**.**
Let and be a non-increasing Fiedler vector of which is constant on each cell. Then is indeed strictly decreasing on the cells from left to right.
Proof.
By contradiction, suppose that there are two vertices in two different cells with the same weight under . We may assume that and that at least one of or has a neighbor with . Let and be the sum of the weights of the neighbors of and , respectively. Then, from the structure of the graphs in , it is evident that . But we have the strict inequality by the existence of .
We may suppose that . Let be the second largest eigenvalue of the adjacency matrix of . Then and . We choose a real with . Now, in the vector we replace the weights of and by and , respectively, to obtain a new vector . As , we have . We have
[TABLE]
where the right hand side is larger than by the choice of , a contradiction. ∎
Lemma 3.14**.**
Any proper elementary move on a graph in leaves a graph isomorphic to the original.
Proof.
For the graphs in , with a Fiedler vector which satisfies Lemma 3.13, proper switchings cannot be found except when are in the same cell, and are in the same cell, , , , and . In this case, leaves a graph isomorphic to the original. Also, any proper elementary move on , and , , , and gives a structure isomorphic to themselves. ∎
Now we can settle the ‘second half’ of Theorem 3.1. The following theorem, combined with Theorem 3.11, completes the proof of Theorem 3.1.
Theorem 3.15**.**
Let be a connected quartic graph such that after a sequence of proper switchings, it is turned to . If , then is isomorphic to .
Proof.
Let be a sequence of proper switchings which turn into . Consider the graphs in which is obtained from by applying . Since , we have , for . Let . Then
[TABLE]
It follows that or . Without loss of generality, suppose that . From Lemma 3.13 it then follows that are in the same cell of . Note that is the reverse of , and so, when applied on , yields . However, is indeed a proper switching, and so by Lemma 3.14, must be isomorphic to . Similarly, it follows that all , for , are isomorphic to . ∎
3.3 Concluding Remarks
By Theorem 3.1 it can be seen that the connected quartic graphs on vertices with minimum spectral gap are , and the graph of Figure 18c, respectively. For , we pose the following conjecture on the puniness and the precise structure of the connected quartic graphs with minimum spectral gap. The conjecture suggests that in such graphs all middle blocks are and end blocks are one of the short blocks or the block given in Figure 24.
Conjecture 3.16**.**
The connected quartic graph on vertices with minimum spectral gap is the unique graph described below. Let and be non-negative integers such that . Then consists of middle blocks and each end block is one of , or . If , then both end blocks are . If , then the end blocks are and . If , then both end blocks are . If , then the end blocks are and . Finally, if , then the end blocks are and .
Acknowledgment
The research of the second author was in part supported by a grant from IPM (No. 98050211). The authors would like to thank anonymous referees for constructive comments which led to improvement of the presentation of the paper.
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