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On the Two Largest Q-Eigenvalues of Graphs

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On the Two Largest Q-Eigenvalues of Graphs View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 310 (2010) 2858–2866 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On the two largest Q -eigenvalues of graphs JianFeng Wang a,b, Francesco Belardo c,∗, QiongXiang Huang b, Bojana Borovi¢anin d a Department of Mathematics, Qinghai Normal University, Xining, Qinghai 810008, PR China b College of Mathematics and System Science, Xinjiang University, Urumqi 830046, PR China c Department of Mathematics, University of Messina, 98166 Sant'Agata, Messina, Italy d Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia article info a b s t r a c t Article history: In this paper, we first give an upper bound for the largest signless Laplacian eigenvalue Received 1 February 2010 of a graph and find all the extremal graphs. Secondly, we consider the second-largest Received in revised form 7 May 2010 signless Laplacian eigenvalue and we characterize the connected graphs whose second- Accepted 23 June 2010 largest signless Laplacian eigenvalue does not exceed 3. Furthermore, we give the signless Available online 21 July 2010 Laplacian spectral characterization of the latter graphs. In particular, the well-known friendship graph is proved to be determined by the signless Laplacian spectrum. Keywords: ' 2010 Elsevier B.V. All rights reserved. Signless Laplacian Largest eigenvalue Cospectral graphs Graph index Spectral characterization Friendship graph 1. Introduction All graphs considered here are undirected and simple (i.e., loops and multiple edges are not allowed). Let G D G.V .G/; E.G// be a graph with vertex set V .G/ D fv1; v2; : : : ; vng and edge set E.G/, where n.G/ D jV .G/j D n is the order and m.G/ D jE.G/j D m is the size of G. For a graph G, let M D M.G/ be a corresponding graph matrix defined in a prescribed way. The M-polynomial of G is defined as det(λI − M/, where I is the identity matrix. The M-eigenvalues of G are the eigenvalues of its M-polynomial. The M-spectrum, denoted by SpecM .G/, of G is a multiset consisting of the M- eigenvalues. The M-spectral radius (or M-index) of G is the largest M-eigenvalue of G. Usually M.G/ is one of the following matrices: A.G/, the adjacency matrix, L.G/, the Laplacian matrix, and Q .G/, the signless Laplacian matrix. Cvetkovi¢ and Simi¢ [6–8] recently investigated the theory of Q -spectra of graphs, and they gave some reasons for studying graphs by using Q -spectra being more efficient than studying them using their A-spectra or L-spectra. See [4,5,2,15,14,21] for some recent results on this aspect. In this paper, we proceed to investigate the properties of the Q - spectra of graphs. Graphs with the same spectrum with respect to a graph matrix M are called M-cospectral graphs. A graph G is said to be determined by its M-spectrum (or we say that G is a DMS-graph for short) if there is no other non-isomorphic graph with ∼ the same spectrum, that is, SpecM .H/ D SpecM .G/ implies H D G for any graph H. An M-cospectral mate of G is a graph cospectral with but not isomorphic to G. The problem ``which graphs are determined by their spectrum?'' was posed by Günthard and Primas [11] more than 50 years ago in the context of Hückel’s theory in chemistry. Only in the most recent ∗ Corresponding author. E-mail addresses: [email protected] (J.F. Wang), [email protected] (F. Belardo), [email protected] (Q.X. Huang), [email protected] (B. Borovi¢anin). 0012-365X/$ – see front matter ' 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2010.06.030 J.F. Wang et al. / Discrete Mathematics 310 (2010) 2858–2866 2859 Fig. 1. Semi-edge walks of length 2 starting from v. years have mathematicians devoted their attention to this problem and many papers are now appearing in many journals. For additional remarks and basic results on this topic we refer the readers to the excellent surveys [19,20]. We now introduce some notation. Let G be a graph order n. Recall that Q .G/ D D.G/ C A.G/, where D.G/ D diag.d1; d2;:::; dn/ with di D dG.vi/ being the degree of vertex vi of G .1 ≤ i ≤ n/. Let S.G/ and L.G/ be respectively the subdivision graph and the line graph of G. The A-polynomial and Q -polynomial of G are respectively denoted by φ.G; λ) and '.G; λ). kG denotes the disjoint union of k copies of G. Assuming the non-increasing order, set SpecA.G/ D .t/ fλ1.G/; λ2.G/; : : : ; λn.G/g and SpecQ .G/ D fq1.G/; q2.G/; : : : ; qn.G/g, while µ means that µ is an eigenvalue with multiplicity t. The eigenvalue q1.G/ will be called the Q -index and it will be denoted by κ.G/. Traditionally, we let Pn; Cn; Kn and K1;n−1 respectively denote the path, the cycle, the complete graph and the star of order n. Kr;s denotes the complete bipartite graph on r C s vertices, where r and s denote the number of vertices in the color classes. A cycle C3 of order 3 is − called a triangle. Kn stands for the graph obtained from Kn by deleting an edge. By Ln;p we denote the lollipop graph obtained by appending a cycle Cp to a pendant vertex of a path Pn−p. Let Ta;b;c denote the tree with exactly one vertex v of maximal degree 3 such that Ta;b;c − v D Pa [ Pb [ Pc . Let Ft denote the friendship graph on n D 2t C 1 vertices, that is, the graph consisting of t triangles intersecting in a single vertex. Let Br;s be the butterfly graph that consists of s triangles sharing a common vertex having an additional r pendant vertices; clearly, the friendship graph Ft is a subgraph of Br;t . The friendship graph is well-known for the following friendship theorem with the wonderful proofs in [10,23]: Friendship Theorem. If G is a graph in which any two distinct vertices have exactly one common neighbor, then G has a vertex joined to all others. The rest of the paper is organized as follows. In Section 2 an upper bound for the Q -index is given. In Section 3 an upper bound for the second-largest Q -eigenvalue is obtained. In Section 4 all connected graphs with the second-largest Q -eigenvalue at most 3 are determined, and their spectral characterization are investigated. 2. An upper bound of the Q -index of a graph Cvetkovi¢, Rowlinson and Simi¢ introduced in [5] a very interesting definition of a ``semi-edge walk'' of a graph which is a generalization of the ``walk'' in a graph. Definition 2.1. A semi-edge walk (of length k) in an (undirected) graph G is an alternating sequence v1; e1; v2; e2; : : : ; vk; ek; vkC1 of vertices v1; v2; : : : ; vkC1 and edges e1; e2;:::; ek such that for any i D 1; 2;:::; k the vertices vi and viC1 are end- vertices (not necessarily distinct) of the edge ei. Lemma 2.1 ([5]). Let Q be the signless Laplacian of a graph G. The .i; j/-entry of the matrix Q k is equal to the number of semi-edge walks of length k starting from vertex i and terminating at vertex j. Using Lemma 2.1 from [9] we get the following lemma which can be proved by a method similar to that used to prove Lemma 2.2 in [9]. Lemma 2.2. Let G be a connected graph of order n and Q its signless Laplacian matrix. Let p.x/ be any polynomial and Sv.p.Q // be the row-sum of p.Q / corresponding to the vertex v. Then min Sv.p.Q // ≤ p(κ.G// ≤ max Sv.p.Q //: v2V .G/ v2V .G/ Moreover, equality holds if and only if the row-sums of p.Q / are all equal. 2 2 2 Now we calculate Sv.Q / in a graph G. Although we can obtain the expression for Sv.Q / by expanding Q D .D.G/ C A.G//2, we prefer to obtain the result through Lemma 2.1. The latter fact can help us to understand the relation between the 2 structure of G and the number of semi-edge walks of a given length. Clearly, Sv.Q / is the number of semi-edge walks of length 2 in G originating at v. By u ∼ v, we mean that u is adjacent to v; otherwise u 6∼ v. All semi-edge walks of length 2 are of the following types (cf. Fig. 1): T1: veiuieijaij and veiuieijui .1 ≤ i ≤ d.v/; 1 ≤ j ≤ d.ui/ − 1/. Each kind of semi-edge walk of type T1 counts d.ui/ − 1 P − such walks. So, the total number of T1 walks is 2 u∼v.d.u/ 1/. 2860 J.F. Wang et al. / Discrete Mathematics 310 (2010) 2858–2866 T2: veiuieiv; veiveiui; veiveiv and veiuieiui .1 ≤ i ≤ d.v//. We have d.v/ semi-edge walks of each kind, so the total number of T2 walks is 4d.v/. T3: veivejuj and veivejv .1 ≤ i 6D j ≤ d.v//.
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