# Sum of squares of degrees in a graph

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1 Sum of squares of degrees in a graph Bernardo M. Ábrego 1 Silvia Fernández-Merchant 1 Michael G. Neubauer William Watkins Department of Mathematics California State University, Northridge Nordhoff St, Northridge, CA, , USA {bernardo.abrego, silvia.fernandez, michael.neubauer, February 1, 008, final version Abstract Let G(v, e be the set of all simple graphs with v vertices and e edges and let P (G = d i denote the sum of the squares of the degrees, d 1,..., d v, of the vertices of G. It is known that the maximum value of P (G for G G(v, e occurs at one or both of two special graphs in G(v, e the quasi-star graph or the quasi-complete graph. For each pair (v, e, we determine which of these two graphs has the larger value of P (G. We also determine all pairs (v, e for which the values of P (G are the same for the quasi-star and the quasi-complete graph. In addition to the quasi-star and quasi-complete graphs, we find all other graphs in G(v, e for which the maximum value of P (G is attained. Density questions posed by previous authors are examined. AMS Subject Classification: 05C07, 05C35 Key words: graph, degree sequence, threshold graph, Pell s Equation, partition, density 1 Partially supported by CIMAT. Current address: Centro de Investigación en Matemáticas, A.C., Jalisco S/N, Colonia Valenciana, 3640, Guanajuato, GTO., México. 1

2 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 1 Introduction Let G(v, e be the set of all simple graphs with v vertices and e edges and let P (G = d i denote the sum of the squares of the degrees, d 1,..., d v, of the vertices of G. The purpose of this paper is to finish the solution of an old problem: 1. What is the maximum value of P (G, for a graph G in G(v, e?. For which graphs G in G(v, e is the maximum value of P (G attained? Throughout, we say that a graph G is optimal in G(v, e, if P (G is maximum and we denote this maximum value by max(v, e. These problems were first investigated by Katz [Ka] in 1971 and by R. Ahlswede and G.O.H. Katona [AK] in In his review of the paper by Ahlswede and Katona. P. Erdős [Er] commented that the solution is more difficult than one would expect. Ahlswede and Katona were interested in an equivalent form of the problem: they wanted to find the maximum number of pairs of different edges that have a common vertex. In other words, they want to maximize the number of edges in the line graph L(G as G ranges over G(v, e. That these two formulations of the problem are equivalent follows from an examination of the vertex-edge incidence matrix N for a graph G G(v, e: trace((nn T = P (G + e trace((n T N = trace(a L (G + 4e, where A L (G is the adjacency matrix of the line graph of G. Thus P (G = trace(a L (G + e. (trace(a L (G is the twice the number of edges in the line graph of G. Ahlswede and Katona showed that the maximum value max(v, e is always attained at one or both of two special graphs in G(v, e. They called the first of the two special graphs a quasi-complete graph. The quasi-complete graph in G(v, e has the largest possible complete subgraph K k. Let k, j be the unique integers such that ( k + 1 e = j, where 1 j k. The quasi-complete graph in G(v, e, which is denoted by QC(v, e, is obtained removing the j edges, (k j + 1, k + 1, (k j +, k + 1,..., (k, k + 1, from the complete graph on the k + 1 vertices, 1,,..., k + 1, and adding v k 1 isolated vertices. The other special graph in G(v, e is the quasi-star, which we denote by QS(v, e. This graph has as many dominant vertices as possible. (A dominant vertex is one with maximum degree v 1. Perhaps the easiest way to describe QS(v, e is to say that it is the graph complement of QC(v, e, where e = ( v e.

3 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 3 Define the function C(v, e to be the sum of the squares of the degree sequence of the quasicomplete graph in G(v, e, and define S(v, e to be the sum of the squares of the degree sequence of the quasi-star graph in G(v, e. The value of C(v, e can be computed as follows: Let e = ( k+1 j, with 1 j k. The degree sequence of the quasi-complete graph in G(v, e is Hence d 1 = = d k j = k, d k j+1 = = d k = k 1, d k+1 = k j, d k+ = = d v = 0. C(v, e = j(k 1 + (k jk + (k j. (1 Since QS(v, e is the complement of QC(v, e, it is straightforward to show that S(v, e = C(v, e + (v 1(4e v(v 1 ( from which it follows that, for fixed v, the function S(v, e C(v, e is point-symmetric about the middle of the interval 0 e ( v. In other words, S(v, e C(v, e = ( S(v, e C(v, e. It also follows from Equation ( that QC(v, e is optimal in G(v, e if and only if QS(v, e is optimal in G(v, e. This allows us to restrict our attention to values of e in the interval [0, ( v /] or equivalently the interval [ ( ( v /, v ]. On occasion, we will do so but we will always state results for all values of e. As the midpoint of the range of values for e plays a recurring role in what follows, we denote it by m = m(v = 1 ( v and define k 0 = k 0 (v to be the integer such that ( ( k0 k0 + 1 m <. (3 To state the results of [AK] we need one more notion, that of the distance from ( k 0 to m. Write ( k0 b 0 = b 0 (v = m. We are now ready to summarize the results of [AK]: Theorem [AK] max(v, e is the larger of the two values C(v, e and S(v, e. Theorem 3 [AK] max(v, e = S(v, e if 0 e < m v and max(v, e = C(v, e if m+ v < e ( v Lemma 8 [AK] If b 0 k 0, or v k 0 1 b 0 < k 0, then C(v, e S(v, e for all 0 e m and ( v C(v, e S(v, e for all m e.

4 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 4 If b 0 < k 0 and k 0 + b 0 < v 1, then there exists an R with b 0 R min {v/, k 0 b 0 } such that C(v, e S(v, e for all 0 e m R C(v, e S(v, e for all m R e m C(v, e S(v, e for all m e m + R ( v C(v, e S(v, e for all m + R e. Ahlswede and Katona pose some open questions at the end of [AK]. Some strange numbertheoretic combinatorial questions arise. What is the relative density of the numbers v for which R = 0 [max(v, e = S(v, e for all 0 e < m and max(v, e = C(v, e for all m < e ( v ]? This is the point of departure for our paper. Our first main result, Theorem 1, strengthens Ahlswede and Katona s Theorem ; not only does the maximum value of P (G occur at either the quasi-star or quasi-complete graph in G(v, e, but all optimal graphs in G(v, e are related to the quasi-star or quasi-complete graphs via their so-called diagonal sequence. As a result of their relationship to the quasi-star and quasi-complete graphs, all optimal graphs can be and are described in our second main result, Theorem. Our third main result, Theorem 6, is a refinement of Lemma 8 in [AK]. Theorem 6 characterizes the values of v and e for which S(v, e = C(v, e and gives an explicit expression for the value R in Lemma 8 of [AK]. Finally, the strange number-theoretic combinatorial aspects of the problem, mentioned by Ahlswede and Katona, turn out to be Pell s Equation y x = ±1. Corollary 3 answers the density question posed by Ahlswede and Katona. Before stating the new results, we summarize work on the problem that followed [AK]. The authors of these papers were often unaware of the work in [AK]. A generalization of the problem of maximizing the sum of the squares of the degree sequence was investigated by Katz [Ka] in 1971 and R. Aharoni [Ah] in Katz s problem was to maximize the sum of the elements in A, where A runs over all (0, 1-square matrices of size n with precisely j ones. He found the maxima and the matrices for which the maxima are attained for the special cases where there are k ones or where there are n k ones in the (0, 1-matrix. Aharoni [Ah] extended Katz s results for general j and showed that the maximum is achieved at one of four possible forms for A. If A is a symmetric (0, 1-matrix, with zeros on the diagonal, then A is the adjacency matrix A(G for a graph G. Now let G be a graph in G(v, e. Then the adjacency matrix A(G of G is a v v (0, 1-matrix with e ones. But A(G satisfies two additional restrictions: A(G is symmetric, and all diagonal entries are zero. However, the sum of all entries in A(G is precisely d i (G. Thus our problem is essentially the same as Aharoni s in that both ask for the maximum of the sum of the elements in A. The graph-theory problem simply restricts the set of (0, 1-matrices to those with e ones that are symmetric and have zeros on the diagonal. Olpp, apparently unaware of the work of Ahlswede and Katona, reproved the basic result that max(v, e = max(s(v, e, C(v, e, but his results are stated in the context of two-colorings of a graph. He investigates a question of Goodman [Go1, Go]: maximize the number of monochromatic triangles in a two-coloring of the complete graph with a fixed number of vertices and a fixed number

5 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 5 of red edges. Olpp shows that Goodman s problem is equivalent to finding the two-coloring that maximizes the sum of squares of the red-degrees of the vertices. Of course, a two-coloring of the complete graph on v vertices gives rise to two graphs on v vertices: the graph G whose edges are colored red, and its complement G. So Goodman s problem is to find the maximum value of P (G for G G(v, e. Olpp shows that either the quasi-star or the quasi-complete graph is optimal in G(v, e, but he does not discuss which of the two values S(v, e, C(v, e is larger. He leaves this question unanswered and he does not attempt to identify all optimal graphs in G(v, e. In 1999, Peled, Pedreschi, and Sterbini [PPS] showed that the only possible graphs for which the maximum value is attained are the so-called threshold graphs. The main result in [PPS] is that all optimal graphs are in one of six classes of threshold graphs. They end with the remark, Further questions suggested by this work are the existence and uniqueness of the [graphs in G(v, e] in each class, and the precise optimality conditions. Some of the results we prove below follow from results in [AK]. We reprove them here for two reasons: First, the new proofs introduce techniques that are used to prove the extensions of the results of [AK]. Second, they make the paper self-contained. We will point out at the appropriate places when a result is not new. Statements of the main results.1 Threshold graphs All optimal graphs come from a class of special graphs called threshold graphs. The quasi-star and quasi-complete graphs are just two among the many threshold graphs in G(v, e. The adjacency matrix of a threshold graph has a special form. The upper-triangular part of the adjacency matrix of a threshold graph is left justified and the number of zeros in each row of the upper-triangular part of the adjacency matrix does not decrease. We will show adjacency matrices using + for the main diagonal, an empty circle for the zero entries, and a black dot, for the entries equal to one. For example, the graph G whose adjacency matrix is shown in Figure 1(a is a threshold graph in G(8, 13 with degree sequence (6, 5, 5, 3, 3, 3, 1, 0. By looking at the upper-triangular part of the adjacency matrix, we can associate the distinct partition π = (6, 4, 3 of 13 with the graph. In general, the threshold graph Th(π G(v, e corresponding to a distinct partition π = (a 0, a 1,..., a p of e, all of whose parts are less than v, is the graph with an adjacency matrix whose upper-triangular part is left-justified and contains a s ones in row s. Thus the threshold graphs in G(v, e are in one-to-one correspondence with the set of distinct partitions, Dis(v, e of e with all parts less than v: Dis(v, e = {π = (a 0, a 1,..., a p : v > a 0 > a 1 > > a p > 0, a s = e}

6 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 6 We denote the adjacency matrix of the threshold graph Th(π corresponding to the distinct partition π by Adj(π. Peled, Pedreschi, and Sterbini [PPS] showed that all optimal graphs in a graph class G(v, e must be threshold graphs. Lemma 1 [PPS] If G is an optimal graph in G(v, e, then G is a threshold graph. Thus we can limit the search for optimal graphs to the threshold graphs. Actually, a much larger class of functions, including the power functions, d p dp v for p, on the degrees of a graph are maximized only at threshold graphs. In fact, every Schur convex function of the degrees is maximized only at the threshold graphs. The reason is that the degree sequences of threshold graphs are maximal with respect to the majorization order among all graphical sequences. See [MO] for a discussion of majorization and Schur convex functions and [MR] for a discussion of the degree sequences of threshold graphs.. The diagonal sequence of a threshold graph To state the first main theorem, we must now digress to describe the diagonal sequence of a threshold graph in the graph class G(v, e. Returning to the example in Figure 1(a corresponding to the distinct partition π = (6, 4, 3 Dis(8, 13, we superimpose diagonal lines on the adjacency matrix Adj(π for the threshold graph Th(π as shown in Figure 1(b. (a (b Figure 1: The adjacency matrix, Adj(π, for the threshold graph in G(8, 13 corresponding to the distinct partition π = (6, 4, 3 Dis(8, 13 with diagonal sequence δ(π = (1, 1,,, 3, 3, 1. The number of black dots in the upper triangular part of the adjacency matrix on each of the diagonal lines is called the diagonal sequence of the partition π (or of the threshold graph Th(π.

7 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 7 The diagonal sequence for π is denoted by δ(π and for π = (6, 4, 3 shown in Figure 1, δ(π = (1, 1,,, 3, 3, 1. The value of P (Th(π is determined by the diagonal sequence of π. Lemma Let π be a distinct partition in Dis(v, e with diagonal sequence δ(π = (δ 1,..., δ t. Then P (Th(π is the dot product P (Th(π = δ(π (1,, 3,..., t = t iδ i. i=1 For example, if π = (6, 4, 3 as in Figure 1, then P (Th(π = (1, 1,,, 3, 3, 1 (1,, 3, 4, 5, 6, 7 = 114, which equals the sum of squares of the degree sequence (6, 5, 5, 3, 3, 3, 1 of the graph Th(π. Theorem in [AK] guarantees that one (or both of the graphs QS(v, e, QC(v, e must be optimal in G(v, e. But there may be other optimal graphs in G(v, e, as the next example shows. The quasi-complete graph QC(10, 30, which corresponds to the distinct partition (8, 7, 5, 4, 3,, 1 is optimal in G(10, 30. The threshold graph G, corresponding to the distinct partition (9, 6, 5, 4, 3,, 1 is also optimal in G(10, 30, but is neither quasi-star in G(10, 30 nor quasi-complete in G(v, 30 for any v. The adjacency matrices for these two graphs are shown in Figure. They have the same diagonal sequence δ = (1, 1,,, 3, 3, 4, 4, 4,,, 1, 1 and both are optimal. Figure : Adjacency matrices for two optimal graphs in G(10, 30, QC(10, 30 = Th(8, 7, 5, 4, 3,, 1 and Th(9, 6, 5, 4, 3,, 1, having the same diagonal sequence δ = (1, 1,,, 3, 3, 4, 4, 4,,, 1, 1 We know that either the quasi-star or the quasi-complete graph in G(v, e is optimal and that any threshold graph with the same diagonal sequence as an optimal graph is also optimal. In fact, the converse is also true. Indeed, the relationship between the optimal graphs and the quasi-star and quasi-complete graphs in a graph class G(v, e is described in our first main theorem. Theorem 1 Let G be an optimal graph in G(v, e. Then G = Th(π is a threshold graph for some partition π Dis(v, e and the diagonal sequence δ(π is equal to the diagonal sequence of either the quasi-star graph or the quasi-complete graph in G(v, e.

8 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 8 Theorem 1 is stronger than Lemma 8 [AK] because it characterizes all optimal graphs in G(v, e. In Section.3 we describe all optimal graphs in detail..3 Optimal graphs Every optimal graph in G(v, e is a threshold graph, Th(π, corresponding to a partition π in Dis(v, e. So we extend the terminology and say that the partition π is optimal in Dis(v, e, if its threshold graph Th(π is optimal in G(v, e. We say that the partition π Dis(v, e is the quasi-star partition, if Th(π is the quasi-star graph in G(v, e. Similarly, π Dis(v, e is the quasi-complete partition, if Th(π is the quasi-complete graph in G(v, e. We now describe the quasi-star and quasi-complete partitions in Dis(v, e. First, the quasi-complete graphs. Let v be a positive integer and e an integer such that 0 e ( v. There exists unique integers k and j such that ( k + 1 e = j and 1 j k. The partition π(v, e, qc := (k, k 1,..., j + 1, j 1,..., 1 = (k, k 1,..., ĵ,...,, 1 corresponds to the quasi-complete threshold graph QC(v, e in Dis(v, e. The symbol ĵ means that j is missing. To describe the quasi-star partition π(v, e, qs in Dis(v, e, let k, j be the unique integers such that ( ( v k + 1 e = + j and 1 j k. Then the partition π(v, e, qs = (v 1, v,..., k + 1, j corresponds to the quasi-star graph QS(v, e in Dis(v, e. In general, there may be many partitions with the same diagonal sequence as π(v, e, qc or π(v, e, qs. For example, if (v, e = (14, 8, then π(14, 8, qc = (7, 6, 5, 4, 3,, 1 and all of the partitions in Figure 3 have the same diagonal sequence, δ = (1, 1,,, 3, 3, 4, 3, 3,,, 1, 1. But none of the threshold graphs corresponding to the partitions in Figure 3 is optimal. Indeed, if the quasicomplete graph is optimal in Dis(v, e, then there are at most three partitions in Dis(v, e with the same diagonal sequence as the quasi-complete graph. The same is true for the quasi-star partition. If the quasi-star partition is optimal in Dis(v, e, then there are at most three partition in Dis(v, e having the same diagonal sequence as the quasi-star partition. As a consequence, there are at most six optimal partitions in Dis(v, e and so at most six optimal graphs in G(v, e. Our second main result, Theorem, entails Theorem 1; it describes the optimal partitions in G(v, e in detail.

9 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 9 Figure 3: Four partitions with the same diagonal sequence as π(14, 8, qc Theorem Let v be a positive integer and e an integer such that 0 e ( v. Let k, k, j, j be the unique integers satisfying ( k + 1 e = j, with 1 j k, and e = ( v ( k j, with 1 j k. Then every optimal partition π in Dis(v, e is one of the following six partitions: 1.1 π 1.1 = (v 1, v,..., k + 1, j, the quasi-star partition for e, 1. π 1. = (v 1, v,..., k j 1,..., k 1, if k + 1 k j 1 v 1, 1.3 π 1.3 = (v 1, v,..., k + 1,, 1, if j = 3 and v 4,.1 π.1 = (k, k 1,..., ĵ,...,, 1, the quasi-complete partition for e,. π. = (k j 1, k, k 3,..., 1, if k + 1 k j 1 v 1,.3 π.3 = (k, k 1,..., 3, if j = 3 and v 4. On the other hand, partitions π 1.1 and π.1 always exist and at least one of them is optimal. Furthermore, π 1. and π 1.3 (if they exist have the same diagonal sequence as π 1.1, and if S(v, e C(v, e, then they are all optimal. Similarly, π. and π.3 (if they exist have the same diagonal sequence as π.1, and if S(v, e C(v, e, then they are all optimal. A few words of explanation are in order regarding the notation for the optimal partitions in Theorem. If k = v, then j = v, e = 0, and π 1.1 =. If k = v 1, then e = j v 1, and π 1.1 = (j ; further, if j = 3, then π 1.3 = (, 1. In all other cases k v and then π 1.1, π 1., and π 1.3 are properly defined. If j = k or j = k 1, then both partitions in 1.1 and 1. would be equal to (v 1, v,..., k and (v 1, v,..., k + 1, k 1 respectively. So the condition k + 1 k j 1 merely

10 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 10 ensures that π 1.1 π 1.. A similar remark holds for the partitions in.1 and.. By definition the partitions π 1.1 and π 1.3 are always distinct; the same holds for partitions π.1 and π.3. In general the partitions π i.j described in items and.1-.3 (and their corresponding threshold graphs are all different. All the exceptions are illustrated in Figure 4 and are as follows: For any v, if e {0, 1, } or e {0, 1, } then π 1.1 = π.1. For any v 4, if e = 3 or e = 3, then π 1.3 = π.1 and π 1.1 = π.3. If (v, e = (5, 5 then π 1.1 = π. and π 1. = π.1. Finally, if (v, e = (6, 7 or (7, 1, then π 1. = π.3. Similarly, if (v, e = (6, 8 or (7, 9, then π 1.3 = π.. For v 8 and 4 e ( v 4, all the partitions πi.j are pairwise distinct (when they exist. Figure 4: Instances of pairs (v, e where two partitions π i.j coincide In the next section, we determine the pairs (v, e having a prescribed number of optimal partitions (and hence graphs in G(v, e..4 Pairs (v, e with a prescribed number of optimal partitions. In principle, a given pair (v, e, could have between one and six optimal partitions. It is easy to see that there are infinitely many pairs (v, e with only one optimal partition (either the quasi-star or the quasi-complete. For example the pair (v, ( v only has the quasi-complete partition. Similarly, there are infinitely many pairs with exactly two optimal partitions and this can be achieved in many different ways. For instance if (v, e = (v, v 5 and v 9, then k = v, j = v 4 > 3, and S(v, e > C(v, e (c.f. Corollary. Thus only the partitions π 1.1 and π 1. are optimal. The interesting question is the existence of pairs with 3,4,5, or 6 optimal partitions. Often, both partitions π 1. and π 1.3 in Theorem exist for the same pair (v, e; however it turns out that this almost never happens when they are optimal partitions. More precisely, Theorem 3 If π 1. and π 1.3 are optimal partitions then (v, e = (7, 9 or (9, 18. Similarly, if π. and π.3 are optimal partitions then (v, e = (7, 1 or (9, 18. Furthermore, the pair (9, 18 is the only one with six optimal partitions, there are no pairs with five. If there are more than two optimal partitions for a pair (v, e, then S(v, e = C(v, e, that is, both the quasi-complete and the quasi-star partitions must be optimal. In the next two results, we describe two infinite families of partitions in Dis(v, e, and hence graph

11 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 11 classes G(v, e, for which there are exactly three (four optimal partitions. The fact that they are infinite is proved in Section 9. Theorem 4 Let v > 5 and k be positive integers that satisfy the Pell s Equation (v 3 (k 1 = 1 (4 and let e = ( k. Then (using the notation of Theorem, j = k, k = k + 1, j = k v +, and there are exactly three optimal partitions in Dis(v, e, namely π 1.1 = (v 1, v,..., k +, k v + π 1. = (v, v 3,..., k π.1 = (k 1, k,...,, 1. The partitions π 1.3, π., and π.3 do not exist. Theorem 5 Let v > 9 and k be positive integers that satisfy the Pell s Equation (v 1 (k + 1 = 49 (5 and e = m = 1 ( v. Then (using the notation of Theorem, j = j = 3, k = k, and there are exactly four optimal partitions in Dis(v, e, namely The partitions π 1. and π. do not exist. π 1.1 = (v 1, v,..., k + 1, 3 π 1.3 = (v 1, v,..., k + 1,, 1 π.1 = (k 1, k,..., 4,, 1 π.3 = (k 1, k,..., 4, 3..5 Quasi-star versus quasi-complete In this section, we compare S(v, e and C(v, e. The main result of the section, Theorem 6, is a theorem very much like Lemma 8 of [AK], with the addition that our results give conditions for equality of the two functions. If e = 0, 1,, 3, then S(v, e = C(v, e for all v. Of course, if e = 0, e = 1 and v, or e 3 and v = 3, there is only one graph in the graph class G(v, e. If e = and v 4, then there are two graphs in the graph class G(v, : the path P and the partial matching M, with degree sequences (, 1, 1 and (1, 1, 1, 1, respectively. The path is optimal as P (P = 6 and P (M = 4. But the path is both the quasi-star and the quasi-complete graph in G(v,. If e = 3 and v 4, then the quasi-star graph has degree sequence (3, 1, 1, 1 and the quasi-complete graph is a triangle with degree sequence (,,. Since P (G = 1 for both of these graphs, both are optimal. Similarly, S(v, e = C(v, e for e = ( v j for j = 0, 1,, 3.

12 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 1 Now, we consider the cases where 4 e ( v 4 4. Figures 5, 6, 7, and 8 show the values of the difference S(v, e C(v, e. When the graph is above the horizontal axis, S(v, e is strictly larger than C(v, e and so the quasi-star graph is optimal and the quasi-complete is not optimal. And when the graph is on the horizontal axis, S(v, e = C(v, e and both the quasi-star and the quasi-complete graph are optimal. Since the function S(v, e C(v, e is central symmetric, we shall consider only the values of e from 4 to the midpoint, m, of the interval [0, ( v ]. Figure 5 shows that S(5, e > C(5, e for all values of e: 4 e < m = 150. So, when v = 5, the quasi-star graph is optimal for 0 e < m = 150 and the quasi-complete graph is not optimal. For e = m(5 = 150, the quasi-star and the quasi-complete graphs are both optimal Figure 5: S(5, e C(5, e > 0 for 4 e < m = 150 Figure 6 shows that S(15, e > C(15, e for 4 e < 45 and 45 < e m = 5.5. But S(15, 45 = C(15, 45. So the quasi-star graph is optimal and the quasi-complete graph is not optimal for all 0 e 5 except for e = 45. Both the quasi-star and the quasi-complete graphs are optimal in G(15, Figure 6: S(15, e C(15, e > 0 for 4 e < 45 and for 45 < e m = 5.5 Figure 7 shows that S(17, e > C(17, e for 4 e < 63, S(17, 64 = C(17, 64, S(17, e < C(17, e

13 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 13 for 65 e < m = 68, and S(17, 68 = C(17, Figure 7: S(17, e C(17, e > 0 for 4 e 63 and 65 e < m = 68 Finally, Figure 8 shows that S(3, e > C(3, e for 4 e 119, but S(3, e = C(3, e for 10 e m = Figure 8: S(3, e C(3, e > 0 for 4 e 119, S(3, e = C(3, e for 10 e < m = 16.5 These four examples exhibit the types of behavior of the function S(v, e C(v, e, for fixed v. The main thing that determines this behavior is the quadratic function q 0 (v := 1 ( 1 (k0 3 + (v 5. 4 (The integer k 0 = k 0 (v depends on v. For example, if q 0 (v > 0, then S(v, e C(v, e 0 for all values of e < m. To describe the behavior of S(v, e C(v, e for q 0 (v < 0, we need to define where R 0 = R 0 (v = e 0 = e 0 (v = 8(m e 0 (k 0 1 (k (v 5, ( k0 = m b 0

14 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 14 Our third main theorem is the following: Theorem 6 Let v be a positive integer 1. If q 0 (v > 0, then S(v, e C(v, e for all 0 e m and S(v, e C(v, e for all m e ( v. S(v, e = C(v, e if and only if e, e {0, 1,, 3, m}, or e, e = e 0 and (v 3 (k 0 3 = 1, 7.. If q 0 (v < 0, then C(v, e S(v, e for all 0 e m R 0 C(v, e S(v, e for all m R 0 e m C(v, e S(v, e for all m e m + R 0 C(v, e S(v, e for all m + R 0 e ( v. S(v, e = C(v, e if and only if e, e {0, 1,, 3, m R 0, m} 3. If q 0 (v = 0, then S(v, e C(v, e for all 0 e m and S(v, e C(v, e for all m e ( v. S(v, e = C(v, e if and only if e, e {0, 1,, 3, e 0,..., m} The conditions in Theorem 6 involving the quantity q 0 (v simplify and refine the conditions in [AK] involving k 0 and b 0. The condition b 0 k 0 in Lemma 8 of [AK] can be removed and the result restated in terms of the sign of the quantity k 0 + b 0 (v 1 = q 0 (v. While [AK] considers only the two cases q 0 (v 0 and q 0 (v > 0, we analyze the case q 0 (v = 0 separately. It is apparent from Theorem 6 that S(v, e C(v, e for 0 e m αv if α > 0 is large enough. Indeed, Ahlswede and Katona [AK, Theorem 3] show this for α = 1/, thus establishing an inequality that holds for all values of v regardless of the sign of q 0 (v. We improve this result and show that the inequality holds when α = 1 / Corollary 1 Let α = 1 /. Then S(v, e C(v, e for all 0 e m αv and S(v, e C(v, e for all m + αv e ( v. Furthermore, the constant α cannot be replaced by a smaller value. Theorem 3 in [AK] can be improved in another way. The inequalities are actually strict. Corollary S(v, e > C(v, e for 4 e < m v/ and S(v, e < C(v, e for m+v/ < e ( v 4.

15 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 15.6 Asymptotics and density We now turn to the questions asked in [AK]: What is the relative density of the positive integers v for which max(v, e = S(v, e for 0 e < m? Of course, max(v, e = S(v, e for 0 e m if and only if max(v, e = C(v, e for m e ( v. Corollary 3 Let t be a positive integer and let n(t denote the number of integers v in the interval [1, t] such that max(v, e = S(v, e, for all 0 e m. Then n(t lim = t t.7 Piecewise linearity of S(v, e C(v, e The diagonal sequence for a threshold graph helps explain the behavior of the difference S(v, e C(v, e for fixed v and 0 e ( v. From Figures 5, 6, 7, and 8, we see that S(v, e C(v, e, regarded as a function of e, is piecewise linear and the ends of the intervals on which the function is linear occur at e = ( ( j and e = v ( j for j = 1,,..., v. We prove this fact in Lemma 10. For now, we present an example. Take v = 15, for example. Figure 6 shows linear behavior on the intervals [36, 39], [39, 45], [45, 50], [50, 55], [55, 60], [60, 66], and [66, 69]. There are 14 binomial coefficients ( j for j 15: The complements with respect to ( 15 = 105 are 1, 3, 6, 10, 15, 1, 8, 36, 45, 55, 66, 78, 91, , 10, 99, 95, 90, 84, 77, 69, 60, 50, 39, 7, 14, 0. The union of these two sets of integers coincide with the end points for the intervals on which S(15, e C(15, e is linear. In this case, the function is linear on the 7 intervals with end points: 0, 1, 3, 6, 10, 14, 15, 1, 7, 8, 36, 39, 45, 50, 55, 60, 66, 69, 77, 78, 84, 90, 91, 95, 99, 10, 104, 105. These special values of e correspond to special types of quasi-star and quasi-complete graphs. If e = ( j, then the quasi-complete graph QC(v, e is the sum of a complete graph on j vertices and v j isolated vertices. For example, if v = 15 and j = 9, and e = ( 9 = 36, then the uppertriangular part of the adjacency matrix for QC(15, 1 is shown on the left in Figure 9. And if e = ( ( v j, then the quasi-star graph QS(v, e has j dominant vertices and none of the other v j vertices are adjacent to each other. For example, the lower triangular part of the adjacency matrix for the quasi-star graph with v = 15, j = 1, and e = ( ( 14 1 = 39, is shown on the right in Figure 9.

17 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 17 dot in position (i, j is given by the entry in the following matrix: This follows from the fact that a sum of consecutive odd integers is a square. So to get the sum of squares P (Th(π of the degrees of the threshold graph associated with the distinct partition π, sum the values in the numerical matrix above that occur in the positions with black dots. Of course, an adjacency matrix is symmetric. So if we use only the black dots in the upper triangular part, then we must replace the (i, j-entry in the upper-triangular part of the matrix above with the sum of the (i, j- and the (j, i-entry, which gives the following matrix: E = Thus, P (Th(π = (1,, 3,... δ(π. Lemma is proved.. (6 4 Proofs of Theorems 1 and Theorem 1 is an immediate consequence of Theorem (and Lemmas 1 and. And Theorem can be proved using the following central lemma: Lemma 3 Let π = (v 1, c, c 1,..., ĵ,...,, 1 be an optimal partition in Dis(v, e, where e (v 1 = c j 4 and 1 j c < v. Then j = c and c v 1 so that π = (v 1, c 1, c,...,, 1. We defer the proof of Lemma 3 until Section 5 and proceed now with the proof of Theorem. The proof of Theorem is an induction on v. Let π be an optimal partition in Dis(v, e, then π c is optimal in Dis(v, e. One of the partitions, π, π c contains the part v 1. We may assume without loss of generality that π = (v 1 : µ, where µ is a partition in Dis(v 1, e (v 1. The cases where µ is a decreasing partition of 0, 1,, and 3 will be considered later. For now we shall assume that e (v 1 4.

18 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 18 Since π is optimal, it follows that µ is optimal and hence by the induction hypothesis, µ is one of the following partitions in Dis(v 1, e (v 1: 1.1a µ 1.1 = (v,..., k + 1, j, the quasi-star partition for e (v 1, 1.a µ 1. = (v,..., k j 1,..., k 1, if k + 1 k j 1 v, 1.3a µ 1.3 = (v,..., k + 1,, 1, if j = 3,.1a µ.1 = (k 1, k 1 1,..., ĵ 1,...,, 1, the quasi-complete partition for e (v 1,.a µ. = (k 1 j 1 1, k 1, k 1 3,..., 1, if k k 1 j 1 1 v,.3a µ.3 = (k 1, k 1 1,..., 3, if j 1 = 3, where e (v 1 = k 1 j 1 4, with 1 j 1 k 1. In symbols, π = (v 1, µ i.j, for one of the partitions µ i.j above. For each partition, µ i.j, we will show that (v 1, µ i.j = π s.t for one of the six partitions, π s.t, in the statement of Theorem. The first three cases are obvious: (v 1, µ 1.1 = π 1.1 (v 1, µ 1. = π 1. (v 1, µ 1.3 = π 1.3. Next suppose that µ = µ.1, µ., or µ.3. The partitions µ. and µ.3 do not exist unless certain conditions on k 1, j 1, and v are met. And whenever those conditions are met, the partition µ.1 is also optimal. Thus π 1 = (v 1, µ.1 is optimal. Also, since e (v 1 4, then k 1 3. There are two cases: k 1 = v, k 1 v 3. If k 1 = v, then µ. does not exist and { π.1, if µ = µ (v 1, µ =.1 π 1.1, if µ = µ.3. If k 1 v 3, then by Lemma 3, π 1 = (v 1, k 1 1,...,, 1, with j 1 = k 1 and k 1 v 1. We will show that k = k and v 1 = k j 1. The above inequalities imply that ( k1 + 1 = k 1 e ( k1 + 1 = k 1 + (v 1 ( ( k1 + 1 k1 + < + (k =. But k is the unique integer satisfying ( ( k e < k+1. Thus k = k1 + 1.

20 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 0 a summand, depending on the location of the black dot in the matrix E in (6. For example, if π = (3, 1, then the part of (1/E that corresponds to the black dots in the adjacency matrix Adj(π for π is Adj((3, 1 = + + +, Thus P (Th(π = ( = 18. Now if we index the rows and columns of the adjacency matrix starting with zero instead of one, then the integer appearing in the matrix (1/E at entry (i, j is just i + j. So we can compute P (Th(π by adding all of the positions (i, j corresponding to the positions of black dots in the upper-triangular part of the adjacency matrix of Th(π. What are the positions of the black dots in the adjacency matrix for the threshold graph corresponding to a partition π = (a 0, a 1,..., a p? The positions corresponding to a 0 are and the positions corresponding to a 1 are (0, 1, (0,,..., (0, a 0 (1,, (1, 3..., (1, 1 + a 1. In general, the positions corresponding to a t in π are We use these facts in the proof of Lemma 3. (t, t + 1, (t, t +,..., (t, t + a t. Let µ = (c, c 1,..., ĵ,...,, 1 be the quasi-complete partition in Dis(v, e (v 1, where 1 j c < v and c j 4. We deal with the cases j = 1, j = c, and j c 1 separately. Specifically, we show that if π = (v 1 : µ is optimal, then j = c and with c v 1. Arguments for the cases are given below. π = (v 1, c 1,...,, 1, (7 +

22 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 5. j = c : µ = (c 1,...,, 1 Since (c 1 4, then c 4. We will show that if c v 1, then π has the same diagonal sequence as the quasi-complete partition. And if c < v 1, then π is not optimal. The adjacency matrix for π is of the following form: 0 1 c v c v c v 1 The quasi-complete partition in G(v, e is π = (c + 1, c,..., k,...,, 1, where k = c v +. To see this, notice that c + (c + 1 k = (c 1 + (v 1 for k = c v +. Since c v 1 and c < v, then 1 k < c and π Dis(v, e. To see that π and π have the same diagonal sequence, we again make a list of the positions in which there is a black dot in Adj(π but not in Adj(π (the top row below, and the positions in which there is a black dot in Adj(π but not in Adj(π (the bottom row below: (0, c + (0, c + 3 (0, c + t + 1 (0, v 1 (1, c + 1 (, c + 1 (t, c + 1 (v c, c + 1. Each position in the top row is in the same diagonal as the corresponding position in the bottom row, that is, 0 + (c + t + 1 = t + (c + 1. Thus the diagonal sequences δ(π = δ(π. 5.. c < v 1 In this case, let π = (v 1, c, c 3,..., 3,. And since c v 3, the parts of π are distinct and decreasing. That is, π Dis(v, e.

23 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 3 Using the fact that c, we again list the positions in which there is a black dot in Adj(π but not in Adj(π (the top row below, and the positions in which the is a black dot in Adj(π but not in Adj(π: (, c (3, c (c 1, c (c, c 1 (1, c + 1 (1, c +... (1, c (1, c 1. All of the positions but the last in the top row are on the same diagonal as the corresponding position in the bottom row: t + c = 1 + (c 1 + t. But in the last positions we have (c + (c 1 = c 3 and 1 + (c 1 = c. Thus P (π = P (π + 6 and so π is not optimal < j < c : µ = (c, c 1,..., ĵ,...,, 1 We will show that π = (v 1, c, c 1,..., ĵ,...,, 1 is not optimal. The adjacency matrix for π has the following form: There are two cases. c 1 c c + 1 c + v c j c j c 1 + c + c v c > v 1 Let π = (v r, c, c 1,..., j + 1 r,...,, 1, where r = min(v 1 c, j. Then r > 1 because j > 1 and c < v. We show that π Dis(v, e and P (π > P (π. In order for π to be in Dis(v, e, the sum of the parts in π must equal the sum of the parts in π: c + (v r (j + 1 r = c + (v 1 j. And the parts of π must be distinct and decreasing: v r > c > j + 1 r > 1.

25 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 5 6 Proof of Theorem 6 and Corollaries 1 and The notation in this section changes a little from that used in Section 1. In Section 1, we write e = ( k+1 j, with 1 j k. Here, we let t = k j so that ( k e = + t, (8 with 0 t k 1. Then Equation (1 is equivalent to C(v, e = C(k, t = (k t(k 1 + tk + t = k(k 1 + t + t(k 1. (9 Before proceeding, we should say that the abuse of notation in C(v, e = C(k, t should not cause confusion as it will be clear which set of parameters (v, e vs. (k, t are being used. Also notice that if we were to expand the range of t to 0 t k, that is allow t = k, then the representation of e in Equation (8 is not unique: e = ( k + k = But the value of C(v, e is the same in either case: Thus we may take 0 t k. ( k C(k, k = C(k + 1, 0 = (k + 1k. We begin the proofs now. At the beginning of Section.5, we showed that S(v, e = C(v, e for e = 0, 1,, 3. Also note that, when m is an integer, Diff(v, m = 0. We now compare S(v, e with C(v, e for 4 e < m. The first task is to show that S(v, e > C(v, e for all but a few values of e that are close to m. We start by finding upper and lower bounds on S(v, e and C(v, e. Define U(e = e( 8e and (( k ( U(k, t = + t (k 1 + 8t 1. The first lemma shows that U(e is an upper bound for C(v, e and leads to an upper bound for S(v, e. Arguments used here to obtain upper and lower bounds are similar to those in [Ni]. Lemma 4 For e C(v, e U(e and S(v, e U(e + (v 1(4e v(v 1. It is clearly enough to prove the first inequality. The second one is trivially obtained from Equation ( linking the values of S(v, e and C(v, e.

26 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 6 Proof. We prove the inequality in each interval ( ( k e k+1. So fix k for now. We make yet another change of variables to get rid off the square root in the above expression of U(k, t. Set t(x = (x (k 1 /8, for k 1 x k + 1. Then U(k, t(x C(k, t(x = 1 64 (x (k 1((k + 1 x ( x + 4(k (k + x 1, which is easily seen to be positive for all k and all k 1 x k + 1. Now define L(e = e( 8e and (( k ( L(k, t = + t (k 1 + 8t 1.5. The next lemma shows that L(e is a lower bound for C(v, e and leads to a lower bound for S(v, e. Lemma 5 For e 3 C(v, e L(e and S(v, e L(e + (v 1(4e v(v 1. Proof. As above, set t(x = (x (k 1 /8, k 1 x k + 1, and x(k, b = k + b, 1 b 1. Then C(k, t(x(k, b L(k, t(x(k, b = 1 64 b (b + 4k (4k 7 3 ( b + This expression is easily seen to be positive for k 3. (k + 1 4k(k k 7 64(4k 7 We are now ready to prove that S(v, e > C(v, e for 0 e m for all but a few small values and some values close to m. Lemma 6 Assume v 5. For 4 e < v we have C(v, e < S(v, e. Proof. As we showed above in Lemma 4, e( 8e is an upper bound on C(v, e for all 1 e ( v. Furthermore, it is easy to see that for 1 e < v we have S(v, e = e + e. In fact, the quasi-star graph is optimal for 1 e < v. The rest is then straightforward. For 4 e, we have 0 < (e 3(e 1 = (e + (8e + 1. Taking square roots and rearranging some terms proves the result.

27 Ábrego, Fernández, Neubauer, Watkins, February 1, 008, final version 7 Lemma 7 Assume v 5. For v e m 0.55v we have S(v, e > C(v, e. Proof. Assume that 0 e m. Let e = m d with 0 d m. By Lemmas 4 and 5, we have S(v, e C(v, e L(e + (v 1(4e v(v 1 U(e We focus on the first two terms. Set = (m + d 8(m + d + 1 (m d 8(m d + 1 (( 4(v d + m. h(d = (m + d 8(d + m + 1 (m d 8(m d + 1. (10 By considering a real variable d, it is easy to see that h (d > 0, h ( (0 = 0, and h (3 (d < 0 on the interval in question. Thus h(d is concave down on 0 d m. We are comparing h(d with the line (4(v 1 + 5/d + m/ on the interval [0.55v, m v]. The concavity of h(d allows to check only the end points. For d = m v, we need to check 1 ( v (v 3 4v 1v + 1 8v + 1 > 1 4 v ( 4v 1v + 7 It is messy and elementary to verify this inequality for v 9. For d = 0.55v we need to check ( v v ( v v v +.4v v 6.4v + 1 > v(.35v This inequality holds for v 9. This time the calculations are a bit messier, yet still elementary. For 4 < v 8, we verify the result directly by calculation. In Section 1, we introduced the value e 0 = ( k 0. We now define e 1 = f 1 = f = ( k0 1 ( ( v k0 + 1 ( ( v k0 +. The next lemma shows that those binomial coefficients and their complements are all we need to consider. Lemma 8 e 1, f < m 0.55v.

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