Applications of Eigenvalues in Extremal Graph Theory Olivia Simpson March 14, 201 Abstract In a 2007 paper, Vladimir Nikiforov extends the results of an earlier spectral condition on triangles in graphs. Namely, a graph G of sufficiently large order n whose spectral radius satisfies µ(g) > n 2 /4 contains a cycle of every length t n/. This condition is shown to be sharp and stable. 1 Introduction Until recently, the focus of spectral graph theory has been on applications to graph structure. The biggest results concern fundamental relationships between the spectrum of a graph and its connectedness, bipartitness, and bounds for good cuts per Cheeger ratio. These global properties are important for a number of applications, particularly those concerned with groupings of nodes. However it is useful to identify smaller properties of graphs without explicit search. Results such as those outlined in this essay and in [1], [9], [] emphasize that eigenvalues of graph matrices can also identify the existence of features such as cliques, small cycles, Hamiltonian cycles, and short paths. 1.1 History In the past decade or so, Vladimir Nikiforov has been a pioneer in an ongoing project to prove extremal properties of graphs with spectral methods. Nikiforov has focused specifically on the spectral properties of cliques and cycles in graphs. In 2007, Nikiforov gave a restatement of a result originally due to Nosal in [10] which asserts that a graph with large enough spectral radius must 1
contain a triangle. The following essay outlines the remarkable extension and improvement of this result which guarantees the existence of cycles of every length up to n/, where n is the order of the graph. 2 Nikiforov s result 2.1 Tools and set up For the duration of the report, let G be a graph on n vertices, with n sufficiently large. Let C t and P t be a cycle and a path on t vertices, respectively. We are concerned with the spectrum of the adjacency matrix, and we let µ(g) denote its largest eigenvalue. Additionally we are concerned with δ(g), the minimum degree of G, and k (G), which we let denote the number of triangles in G. We begin with a formal statement of Nikiforov s 2007 result in [4]. Theorem 1. If G is a graph of order n and µ(g) > n 2 /4, then a triangle exists in G. This result will be used to give the following extension, due to Nikiforov in [5]. Theorem 2 ([5], Theorem 1). Let G be a graph of sufficiently large order n with µ(g) > n 2 /4. Then G contains a cycle of length t for every integer t n/. Nikiforov uses the following characterization of the spectral radii of induced subgraphs as a major tool for the proof of Theorem 2. Theorem ([5], Theorem 5). Let 0 < 4α 1, 0 < 2β 1, 1/2 α/4 γ < 1, K 0, and n (42K + 4)/α 2 β. If G is a graph of order n with µ(g) > γn K/n and δ(g) (γ α)n, then there exists an induced subgraph H G with (1 β)n satisfying one of the following conditions: (i) µ(h) > γ(1 + βα/2) (ii) µ(h) > γ and δ(h) > (γ α) The following two versions of facts due to Nikiforov and Schelp in 2006, and Nikiforov and Bollobás in 2007, respectively, will also be useful in the proof of Theorem 2: 2
Fact 1 ([9]). Let G be a non-bipartite graph of sufficiently large order n, and let δ(g) n/. Then C t G for every integer t [4, δ(g) + 1]. Fact 2 ([1]). If G is a graph of order n, then k (G) (µ(g)/n 1/2)n /12. 2.2 Proof of Theorem 2 Proof idea. Let G = n and µ(g) > n 2 /4. Theorem 1 tells us that G contains a triangle, and therefore G cannot be bipartite. If δ(g) > 2/5n, then Fact 1 implies that G contains a cycle of every length t δ(g) + 1 > n/. So assume δ(g) 2n/5. This is where Theorem comes into play. By setting α = 1/10, β = 1/2, γ = 1/2, K = 1, we can conclude that an induced subgraph H G, > n/2 satisfies one of (i) µ(h) > (1/2 + 1/80) (ii) µ(h) > /2, and δ(h) > 2/5 If (ii) holds, we may apply Fact 1 on H similar to the proof of the case when δ(g) > 2n/5. Therefore we address case (i) and demonstrate the power of Theorem. By Fact 2, k (H) ( µ(h) 1 2 The above can be rearranged as k (H) ) 80 12 = 2 vertex u V (H) which is contained in at least k (H) neighborhood of u must therefore have at least 2 result of Erdős and Gallai now comes in handy. > 2. Then there exists a > 2 triangles. The edges. The following Fact ([2]). Any graph on n nodes and more than nk k(k + 1)/2 edges contains a path with more than 2k edges. By setting k = /, it is achieved that the neighborhood of u in H contains a path P t of length t > n/. Finally, the theorem is proved by taking the path P t along with u to construct a cycle for any length up to n/. The impact of this theorem is strengthened by two additional properties. We illustrate the first with an example. Consider the complete bipartite
graph (or the Turán graph over two subsets), G = K n/2, n/2 (= T 2 (n)). In this case, G has no odd cycles, and in fact its spectral radius satisfies µ(g) = µ(k n/2, n/2 ) = n/2 n/2 = n 2 /4. This gives that the result of Theorem 2 is tight. Secondly, Nikiforov proves the stability of this condition. Namely, if G does not contain all cycles of length up to n/ but µ(g) is close to n 2 /4, then G resembles the complete bipartite graph T 2 (n). Impact The results of this paper are a major step in the practice of using spectral methods in extremal graph theory. It was also a huge leap from previous results (namely that graphs of appropriate conditions are simply guaranteed to contain a triangle). It is quite possible to extend these results to guarantee the containment of cycles of lengths longer than n/, and Nikiforov presents this as a possible extension. A perhaps greater impact of this paper, however, is the contribution to the techniques of spectral applications. Nikiforov explains that, beyond major results, a large goal of his practice is to present tools that can be used across a wide range of problems. The focus of this discussion, in particular, will be on the impact and potential of Theorem. The theorem basically gives a way of identifying a well-connected neighborhood in a graph. An alternative statement, due to [8] is that, for a graph with µ(g) large enough but δ(g) not very large, without removing too many low-degree vertices we may obtain a graph H G with either both µ(h) and δ(h) large enough, or µ(h) considerably larger than expected. The application to the result of this essay was to identify a subset with enough edges to guarantee cycles, for example. Nikiforov has used this tool in later proofs as well. An r-joint of size t is a set of t distinct r-cliques sharing an edge, and js r (G) denotes the maximum size of an r-joint in G. In [6], Nikiforov uses Theorem to identify a subgraph H which bounds js r (G) by js r (H). In [7], Nikiforov gives a slight variation of Theorem to prove the existence of paths in graphs, similar to this result on the existence of cycles. These results are unlikely to push the boundaries of modern graph theory, but rather will become widely available tools for reframing graph theoretic results in a simple way. 4
References [1] B. Bollobás, V. Nikiforov, Cliques and the spectral radius, J. Combin. Theory Ser. B., 97, (2007), 859 865. [2] P. Erdős, T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar, 10, (1959), 7 56. [] M. Fiedler, V. Nikiforov, Spectral radius and Hamiltonicity of graphs, Linear Algebra Appl., 42, (2010), 2170 217. [4] V. Nikiforov, Bounds on graph eigenvalues II, Linear Algebra Appl.,427, (2007), 18 189. [5] V. Nikiforov, A spectral condition for odd cycles in graphs, Linear Algebra Appl., 428, (2008), 1492 1498. [6] V. Nikiforov, Spectral saturation: inverting the spectral Turán theorem, the electronic journal of combinatorics 16.R (2009): 1. [7] V. Nikiforov, The spectral radius of graphs without paths and cycles of specified length, Linear Algebra and Its Applications 42.9 (2010), 224 2256. [8] V. Nikiforov, Some new results in extremal graph theory, arxiv preprint arxiv:1107.1121 (2011). [9] V. Nikiforov, R.H. Schelp, Cycle lengths in graphs with large minimum degree, J. Graph Theory, 52, (2006), 157 170. [10] E. Nosal, Eigenvalues of Graphs, Master s thesis, University of Calgary (1970). 5