© Kun-Fu Fang. 2009
Received: 17 February 2009
Accepted: 11 May 2009
Published: 14 May 2009
The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let be the smallest eigenvalue of . In this paper, we have described the -minor free graphs and showed that (A) let be a simple graph with order . If has no -minor, then . (B) Let be a simple connected graph with order . If has no -minor, then , where equality holds if and only if is isomorphic to .
In this paper, all graphs are finite undirected graphs without loops and multiple edges. Let be a graph with vertices, edges, and minimum degree or . The spectral radius of is the largest eigenvalue of its adjacency matrix. Let be the smallest eigenvalue of . The join is the graph obtained from by joining each vertex of to each vertex of . A graph is said to be a minor of if can be obtained from by deleting edges, contracting edges, and deleting isolated vertices. A graph is -minor free if has no -minor.
Brualdi and Hoffman  showed that the spectral radius satisfies , where , with equality if and only if is isomorphic to the disjoint union of the complete graph and isolated vertices. Stanley  improved the above result. Hong et al.  showed that if is a simple connected graph then with equality if and only if is either a regular graph or a bidegreed graph in which each vertex is of degree either or . Hong  showed that if is a -minor free graph then (1) , where equality holds if and only if is isomorphic to ; (2) , where equality holds if and only if is isomorphic to .
The intersection of and is the graph with vertex set and edge set . Suppose is a connected graph and be a minimal separating vertex set of . Then we can write , where and are connected and . Now suppose further that is a complete graph. We say that is a -sum of and , denoted by , if . In particular, let denote a sum of and . Moreover, if or (say ) has a separating vertex set which induces a complete graph, then we can write such that and are connected and is a complete subgraph of . We proceed like this until none of the resulting subgraphs has a complete separating subgraph. The graphs are called the simplical summands of . It is easy to show that the subgraphs are independent of the order in which the decomposition is carried out (see ).
Theorem 2.1 (see , D. W. Hall; K. Wagner).
A graph is said to be a edge-maximal -minor free graph if has no -minor and has at least an -minor, where is obtained from by joining any two nonadjacent vertices of . A graph is called a maximal planar graph if the planarity will be not held by joining any two nonadjacent vertices of .
This follows from Theorem 2.1.
We denote a planar embedding of by still. Since is a maximal planar graph, every face boundary in is a 3-cycle. Hence the outside face boundary in is a 4-cycle, this implies that the graph is not maximal planar.
Further, we have the following results.
Suppose that the graphs are the simplical summands of , namely . By Corollary 2.2, is either a maximal planar graph or a . By Lemma 2.3, there is at most a maximal planar graph in . Hence we have , where , is a maximal planar graph with order .
Lemma 2.5 (see ).
Let be a simple connected edge-maximal -minor free graph with vertices and edges. Suppose that the graphs are the simplical summands of . Then is either a maximal planar graph or the graph by Corollary 2.2. Further, without loss generality, we may assume that is a spanning subgraph of . Let the graph be the intersection of and . Then for . If then is a subgraph of , implies that . If is a maximal planar graph then is a simple planar bipartite graph, implies that by Lemma 2.5. Next we prove this result by induction on . For , . Now we assume it is true for and prove it for . Let and . Then by the induction hypothesis.
Lemma 3.1 (see ).
Let be a simple connected graph with vertices and edges. If ,then , where equality holds if and only if and is either a regular graph or a bidegreed graph in which each vertex is of degree either or .
Applying the properties of the maximal planar graphs, this follows by calculating.
It follows that (1) and (3) are true by Lemma 3.2 and 5(1)( 3). Next we prove that (2) is true too.
Lemma 3.7 (see ).
Lemma 3.8 (see ).
This follows from Lemmas 3.7, 3.8 and Theorem 2.6.
The author wishes to express his thanks to the referee for valuable comments which led to an improved version of the paper. Work supported by NNSF of China (no. 10671074) and NSF of Zhejian Province (no. Y7080364).
- Brualdi RA, Hoffman AJ: On the spectral radius of -matrices. Linear Algebra and Its Applications 1985, 65: 133–146. 10.1016/0024-3795(85)90092-8MathSciNetView ArticleMATHGoogle Scholar
- Stanley RP: A bound on the spectral radius of graphs with edges. Linear Algebra and Its Applications 1987, 87: 267–269. 10.1016/0024-3795(87)90172-8MathSciNetView ArticleMATHGoogle Scholar
- Hong Y, Shu J-L, Fang KF: A sharp upper bound of the spectral radius of graphs. Journal of Combinatorial Theory, Series B 2001,81(2):177–183. 10.1006/jctb.2000.1997MathSciNetView ArticleMATHGoogle Scholar
- Hong Y: Tree-width, clique-minors, and eigenvalues. Discrete Mathematics 2004,274(1–3):281–287.MathSciNetView ArticleMATHGoogle Scholar
- Thomassen C: Embeddings and minors. In Handbook of Combinatorics, Vol. 1, 2. Edited by: Graham R, Grotschel M, Lovasz L. Elsevier, Amsterdam, The Netherlands; 1995:301–349.Google Scholar
- Bondy JA, Murty USR: Graph Theory, Graduate Texts in Mathematics. Volume 244. Springer, New York, NY, USA; 2008:xii+651.MATHGoogle Scholar
- Hong Y, Shu J-L: Sharp lower bounds of the least eigenvalue of planar graphs. Linear Algebra and Its Applications 1999,296(1–3):227–232.MathSciNetView ArticleMATHGoogle Scholar
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