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Mathematics 2019, 7(1), 44; https://doi.org/10.3390/math7010044
The Aα-Spectral Radii of Graphs with Given Connectivity
School of Mathematics and Statistics and Hubei key Laboratory Mathematics Sciences, Central China Normal University, Wuhan 430079, China
Department of Mathematics, Savannah State University, Savannah, GA 31419, USA
Correspondence: [email protected]; Tel.: +1-352-665-3381
These authors contributed equally to this work.
Received: 22 November 2018 / Accepted: 24 December 2018 / Published: 4 January 2019
The -matrix is with , given by Nikiforov in 2017, where is adjacent matrix, and is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of is said to be the -spectral radius of G. In this work, we determine the graphs with largest -spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying -spectral radius are proposed.
Keywords:adjacent matrix; signless Laplacian; spectral radius; connectivity
We consider simple finite connected graph G with the vertex set and the edge set . The number of vertices is the order of a graph, and the number of edges is the size of a graph. Denote the neighborhood of by , and the degree of v by (or briefly ). For and , let or be the number of components of or . or be a vertex(edge) cut set if (or ) and . For , denote the induced subgraph of G, that is, and .
If is adjacency matrix of a graph G, and is its diagonal matrix of the degrees of G, then the signless Laplacian matrix of G is With the successful studies of these matrices, Nikiforov  proposed the -matrixwith . Obviously, is the adjacent matrix and is the half of signless Laplacian matrix of G, respectively. For undefined terminologies and notations, we refer to .
The research of (adjacency, signless Laplacian) spectral radius is an intriguing topic during past decades [3,4,5,6,7,8,9]. For instances, Lovász and J. Pelikán studied the spectral radius of trees . The minimal Laplacian spectral radius of trees with given matching number is given by Feng et al. . The properties of spectra of graphs and their line graphs are studied by Chen . The signless Laplacian spectra of graphs is explored by Cvetković et al. . Zhou  found bounds of signless Laplacian spectral radius and its hamiltonicity. Graphs having none or one signless Laplacian eigenvalue larger than three are obtained by Lin and Zhou . At the same time, the maximal adjacency or signless Laplacian spectral radius have attracted many interests among the mathematical literature including algebra and graph theory. Ye et al.  gave the maximal adjacency or signless Laplacian spectral radius of graphs subject to fixed connectivity.
Inspired by these outcomes, we determine the graphs with largest -spectral radius with given vertex or edge connectivity. In addition, the corresponding extremal graphs are provided and the equations satisfying the -spectral radius are obtained.
In this section, we provide some important concepts and lemmas that will be used in the main proofs.
Denote by G a graph such that is its vertex set and is its edge set. The -matrix of G has the -entry of is if ; if , and otherwise 0. For , let be the eigenvalues of . The -spectral radius of G is considered as the maximal eigenvalue . Let be a real vector of .
By , we have the quadratic formula of can be expressed that
Because is a real symmetric matrix, and by Rayleigh principle, we have the formula
As we know that once X is an eigenvector of for a connected graph G, X should be unique and positive. The corresponding eigenequations for is rewritten as
As , we study the -matrix for below. Based on the definition of -spectral radius, we have
If G is a connected graph, then is a nonnegative irreducible symmetric matrix. By the results of [1,16,17] and adding extra edges to a connected graph, then -spectral radius will increase and the following lemma is straightforward.
If is any proper subgraph of connected graph G, and ρ is the -spectral radius, then
If X is a positive vector and r is a positive number such that , then .
Recall that the vertex connectivity (respectively, edge connectivity) of a graph G is the smallest number of vertices (respectively, edges) such that if we remove them, the graph will be disconnected or be a single vertex. For convenience, let be the set of all graphs of order n, and (respectively, ) be the set of such graphs with order n and vertex (resp., edge) connectivity k. Note that = having some disconnected graphs of order n, and = consisting of the unique graph . Obviously, = = .
Recall the graph obtained from by attaching a vertex together with edges connecting this vertex to q vertices of . is was found by Brualdi and Solehid in terms of stepwise adjacency matrix, but it is Peter Rowlinson who gives the purely combinatorial definition of such graph. For the property of , we refer to [18,19,20]. Clearly, is with an additional isolated vertex. It’s not hard to see that is of vertex (resp., edge) connectivity q. Let be the smallest and largest degrees of vertices in the graph G, respectively.
The graph is the graph in having the largest -spectral radius, and is the graph in or having the smallest -spectral radius.
By Lemma 2, the first statement is clear. For the second one, let G be a graph which attains the maximum -spectral radius in , then G only has two unique connected components: , ; if not, any component of G will be a proper subgraph of . Then , a contradiction. Then this lemma is proved. □
For , is the graph having the largest -spectral radius in .
Denote by G a graph having the largest -spectral radius in . x is a unit (positive) Perron vector of . Let U be the vertex cut of G having k vertices, and these components of be , for . We declare that ; if not, adding all possible edges within the graph , we would get a graph belonging to (because U is the smallest vertex cut set) and with a larger -spectral radius. Similarly, induced subgraph , the subgraphs and are complete subgranph, and every vertex of U connects these vertices of and . Next we prove that one of will be a singleton, which has a unique vertex. If not, suppose that have orders greater than one. Without loss of generality, denote by u a vertex of having a smallest value for x among vertices in . Deleting these edges of incident to u, and connecting all possible edges between and , we get a graph still in . By Lemma 1, , which yields a contradiction. So one of is a singleton, and G is the desired graph . □
For , is the graph having maximum -spectral radius in .
Denote by G a graph having the largest -spectral radius in . x is a unit (positive) Perron vector of . We know that each vertex of G has degree greater than or equal to k. Otherwise . If there is a vertex u in G with degree k, then the edges adjacent to u are an edge cut such that is complete. The statement follows in this case. Then we will suppose that all vertices in G have degrees greater than k. Let be an edge cut set of G having k edges. So consists of only two components , respectively, of order . Obviously are both complete. In addition, neither of is a singleton. Otherwise G would contain a vertex of degree k, which contradicted to the above assumption. So contain more than 1 vertex, i.e., and .
Without loss of generality, suppose that contains a vertex having a minimal value given by x within all vertices of , and consists of vertices such that . Assume that joins t vertices of . Surely .
If , there exist no edges joining and , and otherwise contains a vertex of degree k. Denote by a new graph with vertex set and edge set , where , and , by Lemma 1, we have . Let be another new graph with vertex set and adding all possible edges between and . Note that , and is a proper subgraph of . By Lemma 2, we have . Thus, , a contradiction.
If . Partition the set as: , . Thus, ; .
Let , then since . Note there is vertex since . Let be a new graph having vertex set and edge set , where , and , by Lemma 1, we have . Let be another new graph having vertex set and adding all possible edges between and , adding all edges between and . Note that , and is a proper subgraph of . Lemma 2 implies that . Thus, , a contradiction. The result follows. □
3. Main Results
In this section, we will determine maximizing -spectral radius of of graphs with given connectivity. By Lemma 4 and Lemma 5, we obtain the following Theorem:
The graph is the graph in with -spectral radius, and is the unique one in or with -spectral radius. For , is the graph with maximum -spectral radius in or .
By the Lemmas 3–5, we obtain the results. □
 Given a partition = with , A be any matrix partitioned into blocks , where is an block. Suppose that the block has constant row sums , and let . Then the spectrum of B is contained in the spectrum of A (taking into account the multiplicities of the eigenvalues).
Since contains , we can partition into three different subsets: , in which u is the vertex connecting a complete subgraph with k edges, a subset S is in connecting u, and . Let x be a Perron vector of . and . Note that .
Label the vertices of as with . The maximum eigenvalues of satisfy the equation:
Since the matrix , where D has on the diagonal the vector and A consists of the following three row-vectors, in the order: ; ; . Thus, by the Lemma 6, x is a constant value on the vertex set S, and constant value on the vertex set T. Defining , , also by (1), we get
Then we get
Note that for , that is, . Then we have:
Then we obtain that
Thus, our proof is finished. □
Let G be a graph of order n having vertex/edge connectivity k, where , the maximum adjacency spectral radius is the largest root of the .
By Theorem 2, let , then . It is obvious since . □
By letting the special values for , we have the following corollary.
Let G be a graph of order n having vertex/edge connectivity k, where , the signless Laplacian spectral radius .
By Theorem 2, let , then . It is obvious since . Thus,
Then we get:□
The above result is the same as .
All authors have contributed equally to this work. Investigation and Methodology: C.W.; Methodology and Correction: S.W.
This work was partially supported by the National Natural Science Foundation of China under Grants 11771172 and 11571134.
Conflicts of Interest
The authors declare no conflict of interest.
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