Spectral Bounds for the Connectivity of Regular Graphs with Given Order
The second-largest eigenvalue and second-smallest Laplacian eigenvalue of a graph are measures of its connectivity. These eigenvalues can be used to analyze the robustness, resilience, and synchronizability of networks, and are related to connectivity attributes such as the vertex- and edge-connectivity, isoperimetric number, and characteristic path length. In this paper, we present two upper bounds for the second-largest eigenvalues of regular graphs and multigraphs of a given order which guarantee a desired vertex- or edge-connectivity. The given bounds are in terms of the order and degree of the graphs, and hold with equality for infinite families of graphs. These results answer a question of Mohar.
AMS subject classifications. 05C50, 05C40.
Determining the connectivity of a graph is a problem that arises often in various applications – see for example [kao] and [resilience]. Let and denote the vertex- and edge-connectivity of a connected graph . Let be the Laplacian matrix of , where is the diagonal degree matrix of and is the adjacency matrix of . We denote the eigenvalues of by and the eigenvalues of by . In 1973, Fiedler related the vertex-connectivity of a graph to as follows:
Theorem 1.1 (Fiedler [Fiedler]).
If is a simple, non-complete graph, then .
This seminal result provided researchers with another parameter that quantitatively measures the connectivity of a graph; hence, is known as the algebraic connectivity of . Fiedler’s discovery ignited interest in studying the connectivity of graphs by analyzing the spectral properties of their associated matrices. Akin to other connectivity measures such as vertex-connectivity, edge-connectivity, and isoperimetric number, the algebraic connectivity of a graph has applications in the design of reliable communication networks [liu] and in analyzing the robustness of complex networks [jamakovic1, jamakovic2].
Recall that for a -regular multigraph on vertices, for . Thus, for regular multigraphs, spectral bounds related to connectivity are often expressed in terms of the second-largest eigenvalue, instead of the second-smallest Laplacian eigenvalue.
Below we survey several results relating to . Note that Theorem 1.1 implies , since .
Theorem 1.2 (Chandran [Chandran2004]).
Let be an -vertex -regular simple graph with . Then .
Theorem 1.3 (Krivelevich and Sudakov [K&Sudakovbookchapter]).
Let be a -regular simple graph with . Then .
In 2010, Theorem 1.3 was improved by Cioabă [Cioaba2010] as follows.
Theorem 1.4 (Cioabă [Cioaba2010]).
Let be a nonnegative integer less than , and let be a -regular, simple graph with . Then .
In the same paper, Cioabă also gave improvements of Theorem 1.4 for the following two particular cases.
Theorem 1.5 (Cioabă [Cioaba2010]).
Let be an odd integer and let denote the largest root of . If is a -regular, simple graph such that , then .
The value of above is approximately .
Theorem 1.6 (Cioabă [Cioaba2010]).
Let be any integer. Let be a -regular, simple graph with
The value of above is approximately . Note that Theorems 1.5 and 1.6 are best possible, as there are examples showing that the upper bounds cannot be lowered. The following extension of these results to was conjectured in the Ph.D. thesis of the fourth author [O-thesis] and was resolved in [suil_in_prep].
Theorem 1.7 (O, Park, Park, and Yu [suil_in_prep]).
Let and let be a -regular simple graph with
In 2016, O [O-Edge-connfromeigenvalues] generalized Fiedler’s result to multigraphs, and established similar bounds to those above.
Theorem 1.8 (O [O-Edge-connfromeigenvalues]).
Let be a connected, -regular multigraph with
Theorem 1.9 (O [O-Edge-connfromeigenvalues]).
Let and let be a connected, -regular multigraph. If , then . If is odd and , then .
The results above make assertions about the edge-connectivity of a graph based on its eigenvalues. In more recent papers, Cioabă and Gu [CioabaGu2016] and O [O-algconnmultgraphs] also established analogous results for vertex-connectivity.
Theorem 1.10 (Cioabă and Gu [CioabaGu2016]).
Let be a connected -regular simple graph, , and
Theorem 1.11 (O [O-algconnmultgraphs]).
Let be a -regular multigraph that is not the -vertex -regular multigraph. If , then .
See [abreu, kirkland, rad] and the bibliographies therein for other recent results on algebraic connectivity; see also [dam1, dam2, dam3] for characterizations of the algebraic connectivities of specific families of graphs.
The aim of the present paper is to investigate what upper bounds on the second-largest eigenvalues of regular simple graphs and multigraphs of a given order guarantee a desired vertex-connectivity or edge-connectivity . In other words, we address the following question asked by Mohar (private communication with the fourth author) and alluded to in [CioabaGu2016]:
For a -regular simple graph or multigraph of a given order and for , what is the best upper bound for which guarantees that or that ?
A starting point of our work, which also motivated the above question, comes from Theorem 1.9 [O-Edge-connfromeigenvalues], because despite the fact that the bound was shown to be tight, the tightness comes from the smallest multigraph. This suggests that this bound can be improved, and a natural next step is to look at the case where the number of vertices is fixed. The main results of this work are the following two spectral bounds which guarantee a certain vertex- and edge-connectivity for multigraphs of a given order. We also construct examples which show the bounds are tight.
Let be an -vertex -regular multigraph with and . If , then .
Let be an -vertex -regular multigraph with , where is the second-largest eigenvalue of a certain matrix (see Section 4). Then .
Theorems 1.13 and 1.14 extend the results listed earlier to multigraphs, and improve some of them (e.g. Theorem 1.11). The majority of the related results listed earlier were derived using a variety of combinatorial, linear algebraic, and analytic techniques; moreover, they feature upper bounds for which do not depend on the order of the graph. In contrast, the results derived in the present paper feature bounds for which depend on both the degree and the order of the graphs, and as such are tight for infinite families of graphs. Furthermore, the derivations of these results combine analytic techniques with computer-aided symbolic algebra; this proves to be a powerful approach, easily establishing the desired results in all but finitely-many cases. The remaining cases are verified through a brute-force approach which relies on enumerating all multigraphs with certain properties. In order to avoid enumeration and post-hoc elimination of the exponential number of multigraphs without the desired properties, our approach required the development of novel combinatorial and graph theoretic techniques. While the problem of generating all non-isomorphic simple graphs having a certain degree sequence and other properties is well-studied (cf. [hakimi, hanlon, ruskey]), there are not as many efficiently-implemented algorithms for constrained enumeration of multigraphs (see [taqqu] for some results in this direction). Thus, the developed enumeration procedure may also be of independent interest.
The paper is organized as follows. In the next section, we recall some graph theoretic and linear algebraic notions, specifically those related to eigenvalue interlacing. In Sections 3 and 4, we present our main results. We conclude with some final remarks in Section 5. The Appendix includes further details and computer code for symbolic computations used in some of the proofs.
We note that Theorems 3.1 and 4.1 are not our main results and are not tight, but we include them for completeness since they are general bounds that give a better intuition of the bigger picture. Also, note that the results for simple graphs discussed in this section are not comparable with our bounds for multigraphs in Theorems 1.13 and 1.14. See for instance the upper bound on in Theorem 1.6 [Cioaba2010], which for behaves approximately as , and the upper bound in Theorem 1.9 [O-Edge-connfromeigenvalues], which for behaves as a small constant. Hence, there is a large gap between the upper bounds on the second largest eigenvalue in simple graphs and the upper bounds for multigraphs, which suggests that there may well be room for improvement.
In this paper, a multigraph refers to a graph with multiple edges but no loops; a simple graph refers to a graph with no multiple edges or loops. The order and size of a multigraph are denoted by and , respectively. A double edge (respectively triple edge) in a multigraph is an edge of multiplicity two (respectively three). The degree of a vertex of , denoted , is the number of edges incident to . The degree sequence of is a list of the vertex degrees of . We may abbreviate the degree sequence of by only writing distinct degrees, with the number of vertices realizing each degree in superscript. For example, if is the star graph on vertices, the degree sequence of may be written as .
A vertex cut (respectively edge cut) of is a set of vertices (respectively edges) which, when removed, increases the number of connected components in . A multigraph with more than vertices is said to be -vertex-connected if there is no vertex cut of size . The vertex-connectivity of , denoted , is the maximum such that is -vertex-connected. Similarly, is -edge-connected if there is no edge cut of size ; the edge-connectivity of , denoted , is the maximum such that is -edge-connected. A cut-vertex (respectively cut-edge) is a vertex cut (respectively edge cut) of size one.
Given sets , denotes the number of edges with one endpoint in and the other in . The induced subgraph is the subgraph of whose vertex set is and whose edge set consists of all edges of which have both endpoints in . A matching is a set of edges of which have no common endpoints; a -matching is a matching containing edges. denotes the graph , and denotes the graph . The complete graph on vertices is denoted . An odd path (respectively even path) in a graph is a connected component which is a path with an odd (respectively even) number of vertices. For other graph theoretic terminology and definitions, we refer the reader to [west].
The adjacency matrix of will be denoted by ; recall that in a multigraph, the entry is the number of edges between vertices and . The eigenvalues of are the eigenvalues of its adjacency matrix, and are denoted by . The Laplacian matrix of is equal to , where is the diagonal matrix whose entry is the degree of vertex . The Laplacian eigenvalues of are the eigenvalues of its Laplacian matrix and are denoted by . The dependence of these parameters on may be omitted when it is clear from the context. Let be an matrix; is a principal submatrix of if is a square matrix obtained by removing certain rows and columns of .
A technical tool used in this paper is eigenvalue interlacing (for more details see Section 2.5 of [BH]). Given two sequences of real numbers and with , we say that the second sequence interlaces the first sequence whenever for .
[Interlacing Theorem,[BH]] If is a real symmetric matrix and is a principal submatrix of of order with , then for , , i.e., the eigenvalues of interlace the eigenvalues of .
Let be a partition of the vertex set of a multigraph into non-empty subsets. The quotient matrix corresponding to is the matrix whose entry () is the average number of incident edges in of the vertices in . More precisely, if , and . Note that for a simple graph, is just the average number of neighbors between vertices in and vertices in .
[Corollary 2.5.4, [BH]] The eigenvalues of any quotient matrix interlace the eigenvalues of .
3 Bounds for to guarantee
In this section, we establish an upper bound for the second-largest eigenvalue of an -vertex -regular simple graph or multigraph which guarantees a certain vertex-connectivity. To our knowledge, this is the first spectral bound on the vertex-connectivity of a regular graph which depends on both the degree and the order of the graph.
Let be an -vertex -regular simple graph or multigraph, which is not obtained by duplicating edges in a complete graph on at most vertices; let
where . If , then .
Assume to the contrary that . If is disconnected, then , a contradiction. Now, assume that . Hence, there exists a vertex cut of with . Let be a union of some components of such that , where and . See Figure 1 for an illustration of this partition.
Let , and ; then, we have , and , so the quotient matrix for the partition is
and the characteristic polynomial of with respect to is . Then by Corollary 2.2, we have
We now consider two cases based on whether is a simple graph or a multigraph.
- Case 1:
is a simple graph. If , then , and since the degree of each vertex in is , it holds that . If , is a complete subgraph of , so the vertex in has degree greater than because ; this is a contradiction. Thus and . Moreover, since , it follows that , and hence . Using this inequality, we have
as desired. If , by the same argument as above, we must have , , , and so , as desired.
- Case 2:
is a multigraph. If , then , , . Moreover, since , it follows that and hence . Using this inequality, we have
as desired. If , then , , , and by a similar reasoning as above, , as desired. ∎
3.2 Improved bound for to guarantee
We now improve the result of Theorem 3.1 for the case when is a multigraph and . Recall that in this case, Theorem 3.1 states that if , then . Moreover, in Observation 3.3 it is shown that the following bound from Theorem 3.2 is tight. As discussed in Section 1, the bound of Theorem 3.2 is incomparable with bounds on guaranteeing a certain vertex connectivity for simple graphs (e.g. Theorem 1.10); however, it does improve the bound of Theorem 1.11 for multigraphs.
Let be an -vertex -regular multigraph with and . If , then .
Assume to the contrary that . If , then , a contradiction. Thus, we can assume henceforth that .
Let be a cut-vertex of , and and be two components of with and . Let and ; without loss of generality, we can assume that , and hence that (otherwise the roles of and can be reversed); note that since , we must have ; moreover, . See Figure 2 for an illustration of this partition in the case when .
The quotient matrix for the partition is
and its characteristic polynomial with respect to is
Then by Corollary 2.2, we have , where is the second-largest root of the characteristic polynomial of ; it can be verified that can be expressed as follows:
If we set the derivative of with respect to equal to zero and solve for , we obtain
Finally, when we substitute for , the resulting expression has a minimum at , for , , and , with minimal value . This minimization and some of the algebraic manipulations described above were carried out using symbolic computation in Mathematica; for details, see the Appendix. ∎
Let be a multigraph with the following adjacency matrix:
Then . Moreover, is a -regular multigraph with 5 vertices, , , and . Thus, the bound in Theorem 3.2 is the best possible for this infinite family of multigraphs.
4 Bounds for to guarantee
In this section, we first give an upper bound for in an -vertex -regular multigraph which guarantees that ; its proof is omitted, since it is similar to that of Theorem 3.1. Theorem 4.1 extends a result of Cioabă [Cioaba2010] to multigraphs.
Let be an -vertex -regular multigraph, which is not obtained by duplicating edges in a complete graph on at most vertices. Let
where . If , then .
Now, we will improve the bound in Theorem 4.1 for the case of ; see Observation LABEL:obs_improvement for an explanation of why Theorem 4.2 is an improvement. In Observation LABEL:theo4.2tight, it is shown that the bound in Theorem 4.2 is tight.
Let be an -vertex -regular multigraph with , where is the second-largest eigenvalue of the following matrix: