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Tilburg University Graphs with constant and van Dam, E.R.; Haemers, W.H. Publication date: 1995 Link to publication in Tilburg University Research Portal Citation for published version (APA): van Dam, E. R., & Haemers, W. H. (1995). Graphs with constant μ and μ. (Research memorandum / Tilburg University, Faculty of Economics and Business Administration; Vol. FEW 688). Unknown Publisher. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 01. okt. 2021 Graphs with constant µ and µ EDWIN R. VAN DAM WILLEM H. HAEMERS Tilburg University, Department of Econometrics, P.O. Box 90153, 5000 LE Tilburg, The Netherlands Abstract. A graph G has constant µ = µ(G) if any two vertices that are not adjacent have µ common neighbours. G has constant µ and µ if G has constant µ = µ(G), and its complement G has constant µ = µ(G). If such a graph is regular, then it is strongly regular, otherwise precisely two vertex degrees occur. We shall prove that a graph has constant µ and µ if and only if it has two distinct restricted Laplace eigenvalues. Bruck-Ryser type conditions are found. Several constructions are given and characterized. A list of feasible parameter sets for graphs with at most 40 vertices is generated. 1. Introduction We say that a noncomplete graph G has constant µ = µ(G) if any two vertices that are not adjacent have µ common neighbours. A graph G has constant µ and µ if G has constant µ = µ(G), and its complement G has constant µ = µ(G). It turns out that only two vertex degrees can occur. Moreover, we shall prove that a graph has constant µ and µ if and only if it has two distinct restricted Laplace eigenvalues. The Laplace eigenvalues of a graph are the eigenvalues of its Laplace matrix. This is a square matrix Q indexed by the vertices, with Qxx = kx, the degree of x, Qxy =−1ifxand y are adjacent, and Qxy =0ifx and y are not adjacent. Note that if G has v vertices and Laplace matrix Q, then its complement G has Laplace matrix vI − J − Q. Since the Laplace matrix has row sums zero, it has an eigenvalue 0 with the all-one vector as eigenvector. The eigenvalues with eigenvectors orthogonal to the all-one vector are called restricted. The restricted multiplicity of an eigenvalue is the dimension of the eigenspace orthogonal to the all-one vector. Note that the graphs with one restricted Laplace eigenvalue are the complete and the empty graphs. Graphs with constant µ and µ form a common generalization of two known families of graphs. The regular ones are precisely the strongly regular graphs and for µ = 1 we have the (nontrivial) geodetic graphs of diameter two. Some similarities with so-called neighbourhood-regular or Γ∆-regular graphs (see [4, 7]) occur. These graphs can be defined as graphs G with constant λ and λ, that is, in G any two adjacent vertices have λ common neighbours, and in G any two adjacent vertices have λ common neighbours. Here also only two vertex degrees can occur, but there is no easy 1 algebraic characterization. 2. Laplace eigenvalues and vertex degrees In this section we shall derive some basic properties of graphs with constant µ and µ.We start with an algebraic characterization. THEOREM 2.1. Let G be a graph on v vertices. Then G has constant µ and µ if and only if θ θ G has two distinct restricted Laplace eigenvalues 1 and 2. If so then only two vertex θ θ degrees k1 and k2 can occur, and 1 + 2 = k1 + k2 +1=µ+v−µ and θ θ 1 2 =k1k2 +µ=µv. Proof. Let G have Laplace matrix Q. Suppose that G has two distinct restricted Laplace θ θ θ θ θ θ 1 v−1 eigenvalues 1 and 2. Then (Q − 1I)(Q − 2I) has spectrum {[ 1 2] , [0] } and row θ θ θ θ θθ sums 1 2, so it follows that (Q − 1I)(Q − 2I)=( 1 2/v)J.Ifxis not adjacent to y,so 2 θ θ θ θ Qxy = 0 then Q xy = 1 2/v, and so µ = 1 2/v is constant. Since the complement of G has θ θ distinct restricted Laplace eigenvalues v − 1 and v − 2, it follows that θ θ µ =(v− 1)(v − 2)/v is also constant. 2 Now suppose that µ and µ are constant. If x and y are adjacent then (vI − J − Q) xy = µ, 2 2 2 so µ =(vI+vJ + Q −2vJ −2vQ)xy = Q xy + v, and if x and y are not adjacent, then 2 2 2 Q xy = µ. Furthermore Q xx = kx + kx. Now 2 2 Q =(µ−v)(diag(kx)−Q)+µ(J−I− diag(kx)+Q) + diag(kx + kx) 2 =(µ+v−µ)Q+ diag(kx − kx(µ + v − µ −1)−µ)+µJ. 2 2 Since Q and Q have row sums zero, it follows that kx − kx(µ + v − µ −1)−µ+µv =0 2 θ θ for every vertex x.SoQ−(µ+v−µ)Q+µvI = µJ. Now let 1 and 2 be such that θ θ θ θ θ θ θθ 1 + 2 = µ + v − µ and 1 2 = µv, then (Q − 1I)(Q − 2I)=( 1 2/v)J,soGhas distinct θ θ restricted Laplace eigenvalues 1 and 2. As a side result we obtained that all vertex degrees kx satisfy the same quadratic equation, thus kx can only take two values k1 and k2, and the formulas readily follow. Note that if the restricted Laplace eigenvalues are not integral, then they have multiplicities m1 = m2 =(v− 1)/2. If the Laplace eigenvalues are integral, then their multiplicities are not necessarily fixed by v, µ and µ. For example, there are graphs on 16 vertices with constant µ = 2 and µ = 6 with Laplace spectrum {[8]m, [4]15−m, [0]1} for m =5,6,7,8and9. The following lemma implies that the numbers of vertices of the respective degrees follow from the Laplace spectrum. 2 LEMMA 2.2. Let G be a graph on v vertices with two distinct restricted Laplace θ θ eigenvalues 1 and 2 with restricted multiplicities m1 and m2, respectively. Suppose there are n1 vertices of degree k1 and n2 vertices of degree k2. Then m1 + m2 +1=n1 +n2 =v θ θ and m1 1 + m2 2 = n1k1 + n2k2. Proof. The first equation is trivial, the second follows from the trace of the Laplace matrix. The number of common neighbours of two adjacent vertices is in general not constant, but depends on the degrees of the vertices. LEMMA 2.3. Let G be a graph with constant µ and µ, and vertex degrees k1 and k2. λ Suppose x and y are two adjacent vertices. Then the number of common neighbours xy of x and y equals λ µ 1 k k if x and y both have degree k , 11 1 2 1 λ λ µ 1 if x and y have different degrees, xy 12 λ 22 µ 1 k2 k1 if x and y both have degree k2. Proof. Suppose x and y have vertex degrees kx and ky, respectively. The number of vertices that are not adjacent to both x and y equals µ. The number of vertices adjacent to x but λ not to y equals kx −1− xy, and the number of vertices adjacent to y but not to x equals λ λ λ λ ky −1− xy. Now we have that v =2+ xy + µ + kx −1− xy + ky −1− xy. Thus λ xy = µ − v + kx + ky. By using that k1 + k2 = µ + v − µ − 1, the result follows. Both Theorem 2.1 and Lemma 2.3 imply the following. COROLLARY 2.4. A graph with constant µ and µ is regular if and only if it is strongly regular. 2 Observe that G is regular if and only if (µ + v − µ −1) =4µ(v−1)orn1 =0orn2 =0. Since we can express all parameters in terms of the Laplace spectrum, it follows that it can be recognized from the Laplace spectrum whether a graph is strongly regular or not. This is no surprise, since it is in general true that regularity of a graph follows from its Laplace spectrum. Before proving the next lemma we first look at the disconnected graphs. Since the number of components of a graph equals the multiplicity of its Laplace eigenvalue 0, a graph with constant µ and µ is disconnected if and only if one of its restricted Laplace eigenvalues equals 0. Consequently this is the case if and only if µ =0.Soina disconnected graph G with constant µ and µ any two vertices that are not adjacent have no common neighbours.

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