Sufficient conditions for graphs to be $k$-connected, maximally connected and super-connected

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$) if every minimum vertex-cut isolates a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is $k$-connected, maximally connected or super-connected if the number of edges or the spectral radius is large enough.


Introduction
Let G � (V, E) be a simple connected undirected graph, where V � V(G) is the vertex-set of G and E � E(G) is the edge-set of G. e order and size of G are defined by n � |V(G)| and m � |E(G)|, respectively; d G (x) is the degree of a vertex x in G, that is, the number of edges incident with x in G; δ(G) � min d G (x): x ∈ V(G) is the minimum degree of G. For a subset X ⊂ V(G), use G[X] to denote the subgraph of G induced by X. For two subsets X and Y of V(G), let [X, Y] be the set of edges between X and Y. e complement of G is denoted by G. Let G 1 ∪ G 2 denote the disjoint union of graphs G 1 and G 2 , and let G 1 ∨ G 2 denote the graph obtained from G 1 ∪ G 2 by joining each vertex of G 1 to each vertex of G 2 . e graph G is called a triangle-free graph if G contains no triangle. Denote by ρ(G) the largest eigenvalue or the spectral radius of the adjacency matrix of G and it is called the spectral radius of G. If G is connected, then, by Perron-Frobenius eorem, ρ(G) is simple and there exists a unique (up to a multiple) corresponding positive eigenvector.
A vertex-cut of a connected graph G is a set of vertices whose removal disconnects G. e vertex-connectivity or simply the connectivity κ � κ(G) of a connected graph G is the minimum cardinality of a vertex-cut of G if G is not complete, and κ(G) � n − 1 if G is the complete graph K n of order n. A vertex-cut S is a minimum vertex-cut or a κ-cut of G if |S| � κ(G). Apparently, κ(G) ≤ δ(G) for any graph G. e graph G is k-connected if κ(G) ≥ k, maximally connected if κ(G) � δ(G), and super-connected (or superκ) 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 G is a set of edges whose removal disconnects G. e edge connectivity λ � λ(G) of a connected graph G is defined as the minimum cardinality of an edge-cut over all edge-cuts of G. An edge-cut S is a minimum edge-cut if |S| � λ(G). e inequality λ(G) ≤ δ(G) is obvious. e graph G is maximally edge-connected if λ(G) � δ(G), and it is super-edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. erefore, 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. e 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. Cioabȃ [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 vertexconnectivity, Li [14] presented sufficient conditions for a graph to be k-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 k-connected and k-edge-connected; Feng et al. [16] obtained a tight sufficient condition for a connected graph with fixed minimum degree to be k-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], Cioabȃ and Gu [18], O [19], and Zhang [20]. e relationship between vertexconnectivity and adjacency eigenvalues or Laplacian eigenvalues of graphs has been investigated by Hong et al. [21][22][23] 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 k-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 trianglefree graph with given minimum degree to be k-connected, maximally connected, or super-connected in terms of the number of edges or its spectral radius, respectively. e results on k-connected graph in this paper improve the result in [16] by Feng et al. to some extent. e rest of this paper is organized as follows. In Section 2, we present sufficient conditions for a graph with given minimum degree to be k-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 k � δ, 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 trianglefree graph to be k-connected, maximally connected, or super-connected are acquired in terms of the number of edges and the spectral radius of the graph, respectively.

k-Connected Graphs
Let G be a connected graph of order n, minimum degree δ, and vertex-connectivity κ. If n ≤ 4 or δ � 1, then κ � δ. If δ � n − 1, then G � K n and κ � δ. If δ � n − 2, then when u and v are nonadjacent, the other n − 2 vertices are all common neighbors of u and v. It is necessary to delete all common neighbors of some pair of vertices to separate the graph, so κ ≥ n − 2 � δ. erefore, we only need to consider n ≥ 5 and 2 ≤ δ ≤ n − 3 in the following.

Theorem 3. Let G be a connected graph of order n and minimum degree δ
. , x n ) T be the unique positive unit eigenvector corresponding to ρ(G). Recall that Rayleigh's principle implies that Assume that G is a proper subgraph of H. Clearly, we could assume that G is obtained by omitting just one edge uv of H. Let X, Y, Z be the set of vertices of H of degree δ, n − 1, n − δ + k − 3, respectively, where |X| � δ − k + 2, |Y| � k − 1, and |Z| � n − δ − 1. Since δ(G) � δ, G must contain all the edges between X and Y. erefore, 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, u, v { } ⊂ Y. Choose a vertex w ∈ Z. If x u ≥ x w , then by removing the edge vw and adding the edge uv we obtain a new graph G 1 which is covered by case (b). By the Rayleigh principle, If x w > x u , then by removing all the edges between X and u { } and adding all the edges between X and w { } we obtain a new graph G 1 ′ which is also covered by case (b). By the Rayleigh principle, Secondly, suppose that case (b) occurs; that is, u ∈ Y, v ∈ Z. Choose a vertex w ∈ Z and w ≠ v. If x u ≥ x w , then by removing the edge vw and adding the edge uv we obtain a new graph G 2 which is covered by case (c). By the Rayleigh principle,

(24)
If x w > x u , then by removing all the edges between X and u { } and adding all the edges between X and w { } we obtain a new graph G 2 ′ which is also covered by case (c). By the Rayleigh principle, erefore, we could assume that u, v { } ∈ Z. By symmetry, let x ≔ x i for any i ∈ X; y ≔ x j for any j ∈ Y; z ≔ x ℓ for any ℓ ∈ Z, u, v { }; and t ≔ x u � x v . According to λx i � ij∈E(G) x j and the uniqueness of x, we have that ρ is the largest root of following equations: us, ρ(G) is the largest root of the equation By some basic calculations, we have ). It is easy to see that the function g(x) is strictly increasing when By 3 ≤ k ≤ δ ≤ n − 3 and n ≥ (1/2)(δ − k + 2)(k 2 − 2k+ 7), we have n ≥ 2(δ − k + 3) and then 4 Complexity □ e 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 G be a connected graph with n vertices and m edges. Let δ � δ(G) be the minimum degree of G and let ρ(G) be the spectral radius of the adjacency matrix of G. en, Equality holds if and only if G is either a regular graph or a bidegreed graph in which each vertex is of degree either δ or n − 1.
Another sufficient condition for graphs to be k-connected can be obtained by using the spectral radius of the complement of a graph.

Theorem 5.
Let G be a connected graph of order n ≥ 5 and minimum degree δ ≥ k ≥ 2. If Proof. Let κ � κ(G). Assume that (35) holds but 1 ≤ κ ≤ k − 1. Let S be an arbitrary minimum vertex-cut of G, and let X 0 , X 1 , . . . , X p− 1 (p ≥ 2), denote the vertex-sets of the components of G − S, where |X 0 | ≤ |X 1 | ≤ · · · ≤ |X p− 1 |. Each vertex in X i is adjacent to at most |X i | − 1 vertices of X i and κ � |S| vertices of S. us, Since there are no edges between X 0 and Y in G, K |X 0 |,|Y| is a subgraph of G. us, By (35), the above inequalities must be equalities. us,
Theorem 6. Let G be a connected graph of order n ≥ 5, size m, and minimum degree δ ≥ 2.
Proof. On the contrary, suppose that κ(G) < δ. Since G is connected, by (38) and Lemma 1, we have which yields By eorem 6 (a), . To complete the proof, we only need to show δ � n − 3.
Since |E(G)| � n − 2 2 + 2δ − 1, the equalities hold in (39). us, by Lemma 1, G is either a regular graph or a bidegreed graph in which each vertex is of degree δ or n − 1. However, the vertices of G have degrees from the set δ, n − 3, n − 1 { }. erefore, δ � n − 3 and the result follows.
□ By setting k � δ in eorem 2, we obtain the following result.
Theorem 8. Let G be a connected graph of order n ≥ 5 and minimum degree δ ≥ 2. If ) is the largest root of the equation Theorem 9. Let G be a connected graph of order n and minimum degree δ ≥ 2. If n ≥ δ 2 − 2δ + 7 and then G is maximally connected, unless G � K δ− 1 ∨ (K 2 ∪ K n− δ− 1 ).
By setting k � δ in eorem 5, we have the following result.

Super-Connected Graphs
For any connected graph G of order n, if 2 ≤ n ≤ 4, then G is super-κ. erefore, n ≥ 5 is considered in this section.