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Volume 2021 |Article ID 5588146 | https://doi.org/10.1155/2021/5588146

Zhen-Mu Hong, Zheng-Jiang Xia, Fuyuan Chen, Lutz Volkmann, "Sufficient Conditions for Graphs to Be -Connected, Maximally Connected, and Super-Connected", Complexity, vol. 2021, Article ID 5588146, 11 pages, 2021. https://doi.org/10.1155/2021/5588146

Sufficient Conditions for Graphs to Be -Connected, Maximally Connected, and Super-Connected

Academic Editor: M. Irfan Uddin
Received07 Jan 2021
Revised24 Jan 2021
Accepted27 Jan 2021
Published22 Feb 2021

Abstract

Let be a connected graph with minimum degree and vertex-connectivity . The graph is -connected if , maximally connected if , and super-connected if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we present sufficient conditions for a graph with given minimum degree to be -connected, maximally connected, or super-connected in terms of the number of edges, the spectral radius of the graph, and its complement, respectively. Analogous results for triangle-free graphs with given minimum degree to be -connected, maximally connected, or super-connected are also presented.

1. Introduction

Let be a simple connected undirected graph, where is the vertex-set of and is the edge-set of . The order and size of are defined by and , respectively; is the degree of a vertex in , that is, the number of edges incident with in ; is the minimum degree of . For a subset , use to denote the subgraph of induced by . For two subsets and of , let be the set of edges between and . The complement of is denoted by . Let denote the disjoint union of graphs and , and let denote the graph obtained from by joining each vertex of to each vertex of . The graph is called a triangle-free graph if contains no triangle. Denote by the largest eigenvalue or the spectral radius of the adjacency matrix of and it is called the spectral radius of . If is connected, then, by Perron-Frobenius Theorem, is simple and there exists a unique (up to a multiple) corresponding positive eigenvector.

A vertex-cut of a connected graph is a set of vertices whose removal disconnects . The vertex-connectivity or simply the connectivity of a connected graph is the minimum cardinality of a vertex-cut of if is not complete, and if is the complete graph of order . A vertex-cut is a minimum vertex-cut or a -cut of if . Apparently, for any graph . The graph is -connected if , maximally connected if , and super-connected (or super-) if every minimum vertex-cut isolates a vertex of minimum degree. Hence, every super-connected graph is also maximally connected. An edge-cut of a connected graph is a set of edges whose removal disconnects . The edge connectivity of a connected graph is defined as the minimum cardinality of an edge-cut over all edge-cuts of . An edge-cut is a minimum edge-cut if . The inequality is obvious. The graph is maximally edge-connected if , and it is super-edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. Therefore, every super-edge-connected graph is also maximally edge-connected. For graph-theoretical terminology and notation not defined here, one can refer to [1, 2].

Sufficient conditions for graphs to be maximally (edge-) connected or super-(edge-) connected were given by several authors, depending on the order, the maximum and minimum degree, the diameter, the girth, the degree sequence, the clique number and so on. The paper in [3] by Hellwig and Volkmann gives a survey on this topic. Recently, Volkmann and Hong [4] proved that a connected graph or a connected triangle-free graph is maximally edge-connected or super-edge-connected if the number of edges is large enough, and the results corresponding to triangle-free graphs were generalized to connected graphs with given clique number by Volkmann [5].

On the other hand, the relationship between graph properties and eigenvalues has attracted much attention. Fiedler [6] initiated the research on the relationship between graph connectivity and graph eigenvalues, and Fiedler and Nikiforov [7] initiated the investigation on the spectral conditions for graphs to be Hamiltonian or traceable. Cioabă [8] investigated the relationship between edge-connectivity and adjacency eigenvalues of regular graphs. From then on, the edge-connectivity problem has been intensively studied by many researchers, such as Duan et al. [9], Gu et al. [10], Liu et al. [11], Liu et al. [12], and Suil O [13]. For vertex-connectivity, Li [14] presented sufficient conditions for a graph to be -connected using spectral radius and signless Laplacian spectral radius; Feng et al. [15] demonstrated sufficient conditions based on spectral radius for a graph to be -connected and -edge-connected; Feng et al. [16] obtained a tight sufficient condition for a connected graph with fixed minimum degree to be -connected based on its spectral radius, for sufficiently large order. Vertex-connectivity and the second largest adjacency eigenvalue of regular graphs were studied by Abiad et al. [17], Cioabă and Gu [18], O [19], and Zhang [20]. The relationship between vertex-connectivity and adjacency eigenvalues or Laplacian eigenvalues of graphs has been investigated by Hong et al. [2123] and Liu et al. [24].

Motivated by the researches mentioned above, this paper presents sufficient conditions for a graph with given minimum degree to be -connected, maximally connected, or super-connected in terms of the number of edges, the spectral radius of the graph, or its complement, respectively. In addition, we also give sufficient conditions for a triangle-free graph with given minimum degree to be -connected, maximally connected, or super-connected in terms of the number of edges or its spectral radius, respectively. The results on -connected graph in this paper improve the result in [16] by Feng et al. to some extent.

The rest of this paper is organized as follows. In Section 2, we present sufficient conditions for a graph with given minimum degree to be -connected in terms of the number of edges, the spectral radius of the graph, and its complement, respectively. In terms of the same parameters as in Section 2, by setting , we get sufficient conditions for a graph with given minimum degree to be maximally connected in Section 3, and we obtain sufficient conditions for a graph with given minimum degree to be super-connected in Section 4. In Section 5, sufficient conditions for a triangle-free graph to be -connected, maximally connected, or super-connected are acquired in terms of the number of edges and the spectral radius of the graph, respectively.

2. -Connected Graphs

Let be a connected graph of order , minimum degree , and vertex-connectivity . If or , then . If , then and . If , then when and are nonadjacent, the other vertices are all common neighbors of and . It is necessary to delete all common neighbors of some pair of vertices to separate the graph, so . Therefore, we only need to consider and in the following.

Theorem 1. Let be a connected graph of order , size , and minimum degree .(a)Ifthen is -connected, unless .(b)If andthenis-connected, unlessis a subgraph of.

Proof. Let . On the contrary, suppose that is not -connected; that is, . Let be an arbitrary minimum vertex-cut, and let (), denote the vertex sets of the components of , where . Each vertex in is adjacent to at most vertices of and vertices of . Thus,and so . Let ; then . Therefore,Since is disconnected, there are no edges between and in and(a)Since we suppose that is not -connected, it suffices to prove . By (4) and , and since , we obtainSubstituting (6) into (5), it follows thatCombining this with (1), we obtain . Hence, all the inequalities in (6) must be equalities and so , , and . Thus, is obtained from by deleting all the edges of the complete bipartite subgraph of . That is, , , , and .(b)To prove that is a subgraph of , we first show that . Suppose that . Since , , and , we haveSubstituting (8) into (5), it follows thatCombining this with (2), it is easy to get . By the hypothesis, we have . Hence, and all the inequalities in (8) must be equalities. Thus, , , , and is obtained from by deleting all the edges of the complete bipartite subgraph of . That is, . However, , a contradiction. Thus, . Combining this with , we get . Since and for each , we have that each vertex of is adjacent to each vertex of and . Therefore, and is a subgraph of .

Theorem 2. Let be a connected graph of order and minimum degree . Ifthenis-connected, unless, whereis the largest root of the equation

Proof. Let . Assume that (10) holds but . Let be an arbitrary minimum vertex-cut of , and let (), denote the vertex-sets of the components of , where . Each vertex in is adjacent to at most vertices of and vertices of . Thus,and so for each . Let . Then, and . Since there are no edges between and in , is a subgraph of and .
Next, we will showDenote for short, where and . Let be the unique positive unit eigenvector corresponding to . By symmetry, let for any ; for any ; for any . According to and the uniqueness of , we have that is the largest root of the following equations:Thus, is the largest root of the equationThen, we havefor any and . Therefore, for any , which means thatSince is a subgraph of for any ,Hence, from the discussion above, we haveBy (10), the above inequalities must be equalities. Thus, , , , and so . The result follows from (15).

Remark 1. In Corollary 3.5 in [16], the authors showed that if is a connected graph of minimum degree and order , and , then is -connected, unless . Apparently, without restriction on the order of graph, Theorem 2 improves Corollary 3.5 in [16].

Theorem 3. Let be a connected graph of order and minimum degree . If is a subgraph of and , thenunless.

Proof. Denote for short. Let be the unique positive unit eigenvector corresponding to . Recall that Rayleigh’s principle implies thatAssume that is a proper subgraph of . Clearly, we could assume that is obtained by omitting just one edge of . Let be the set of vertices of of degree , respectively, where , , and . Since , must contain all the edges between and . Therefore, , with three possible cases: (a) ; (b) ; and (c) . We shall show that case (c) yields a graph whose spectral radius is not smaller than the spectral radius of the graph in case (b) and that case (b) yields a graph whose spectral radius is not smaller than the spectral radius of the graph in case (a).
Firstly, suppose that case (a) occurs; that is, . Choose a vertex . If , then by removing the edge and adding the edge we obtain a new graph which is covered by case (b). By the Rayleigh principle,If , then by removing all the edges between and and adding all the edges between and we obtain a new graph which is also covered by case (b). By the Rayleigh principle,Secondly, suppose that case (b) occurs; that is, . Choose a vertex and . If , then by removing the edge and adding the edge we obtain a new graph which is covered by case (c). By the Rayleigh principle,If , then by removing all the edges between and and adding all the edges between and we obtain a new graph which is also covered by case (c). By the Rayleigh principle,Therefore, we could assume that . By symmetry, let for any ; for any ; for any ; and . According to and the uniqueness of , we have that is the largest root of following equations:Thus, is the largest root of the equationBy some basic calculations, we haveSet . It is easy to see that the function is strictly increasing when . Since , we getBy and , we have and then, , and . Therefore, it is easy to find that the largest root of is in the interval , which yields .

The following lemma gives a sharp upper bound of the spectral radius of connected graphs with given number of edges and minimum degree.

Lemma 1 (see [25]). Let be a connected graph with vertices and edges. Let be the minimum degree of and let be the spectral radius of the adjacency matrix of . Then,

Equality holds if and only if is either a regular graph or a bidegreed graph in which each vertex is of degree either or .

Theorem 4. Let be a connected graph of order and minimum degree . If andthen is -connected, unless .

Proof. On the contrary, suppose that . Since is connected and , by Lemma 1, we havewhich yieldsSince , we obtain . By Theorem 1 (b), is a subgraph of . Since , by Theorem 3, . The proof is completed.

Remark 2. In Theorem 3.4 in [16], the authors proved that if is a connected graph of minimum degree and order , and , then is -connected unless . Obviously, Theorem 4 improves Theorem 3.4 in [16] from the perspective of the restriction on the order of graph.

Another sufficient condition for graphs to be -connected can be obtained by using the spectral radius of the complement of a graph.

Theorem 5. Let be a connected graph of order and minimum degree . Ifthenis-connected, unless.

Proof. Let . Assume that (35) holds but . Let be an arbitrary minimum vertex-cut of , and let (), denote the vertex-sets of the components of , where . Each vertex in is adjacent to at most vertices of and vertices of . Thus,and so for each . Let . Then, and . Since there are no edges between and in , is a subgraph of . Thus,By (35), the above inequalities must be equalities. Thus, , and , and so .

3. Maximally Connected Graphs

If , then is maximally connected. Therefore, by setting in Theorem 1, we obtain the following theorem.

Theorem 6. Let be a connected graph of order , size , and minimum degree .(a)If, thenis maximally connected, unless.(b)Ifand, thenis maximally connected, unlessis a subgraph of.

Theorem 7. Let be a connected graph of order and minimum degree . Ifthenis maximally connected, unless.

Proof. On the contrary, suppose that . Since is connected, by (38) and Lemma 1, we havewhich yieldsBy Theorem 6 (a), . To complete the proof, we only need to show .
Since , the equalities hold in (39). Thus, by Lemma 1, is either a regular graph or a bidegreed graph in which each vertex is of degree or . However, the vertices of have degrees from the set . Therefore, and the result follows.

By setting in Theorem 2, we obtain the following result.

Theorem 8. Let be a connected graph of order and minimum degree . Ifthenis maximally connected, unless, whereis the largest root of the equation

Theorem 9. Let be a connected graph of order and minimum degree . If andthen is maximally connected, unless .

Proof. Set in the proofs of Theorems 3 and 4. If , then the result follows from Theorem 4. If , then case (a) cannot occur in the proof of Theorems 3. In Theorem 3, by noting that , , , , and , we have and so Theorem 3 holds for . Hence, Theorem 4 also holds for and the result follows.

Remark 3. In the proof of Theorem 3, if we take and , then when . Notice that . So, the largest root of is greater than if , and it follows that . That is to say, the requirement in Theorem 9 is best possible when .

By setting in Theorem 5, we have the following result.

Theorem 10. Let be a connected graph of order and minimum degree . Ifthen is maximally connected, unless .

4. Super-Connected Graphs

For any connected graph of order , if , then is super-. Therefore, is considered in this section.

Theorem 11. Let be a connected graph of order , size , and minimum degree . Ifthenis super-, unless, whereis an edge ofwithand.

Proof. Since , by Theorem 6 (a), . On the contrary, suppose that is not super-. Let be an arbitrary minimum vertex-cut with vertices, and let () denote the vertex-sets of the components of , where . Denote . Since is disconnected, there are no edges between and in , andThus, by