Determining Finite Connected Graphs Along the Quadratic Embedding Constants of Paths

The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants of paths $P_n$ form a strictly increasing sequence converging to $-1/2$. Then we formulate the problem of determining all the graphs $G$ satisfying $\mathrm{QEC}(P_n)\le\mathrm{QEC}(G)<\mathrm{QEC}(P_{n+1})$. The answer is given for $n=2$ and $n=3$ by exploiting forbidden subgraphs for $\mathrm{QEC}(G)<-1/2$ and the explicit QE constants of star products of the complete graphs.


Introduction
Let G = (V, E) be a finite connected graph with |V| = n ≥ 2 and D = [d(i, j)] i, j∈V the distance matrix of G. The quadratic embedding constant (QE constant for short) of G is defined by QEC(G) = max{ f, D f ; f ∈ C(V), f, f = 1, 1, f = 0}, (1.1) where C(V) is the space of all R-valued functions on V, 1 ∈ C(V) the constant function taking value 1, and ·, · the canonical inner product. The QE constant was first introduced for the quantitative study of quadratic embedding of graphs in Euclidean spaces [20,21]. In particular, a graph G admits a quadratic embedding (in this case we say that G is of QE class) if and only if QEC(G) ≤ 0. Moreover, it is noteworthy that QEC(G) ≤ 0 is equivalent to the positive definiteness of the Qmatrix Q = [q d (i, j) ] for all 0 ≤ q ≤ 1. This property, first proved by Haagerup [9] for trees and later by Bożejko [6] for general star products, has many applications in harmonic analysis and quantum probability, see [5,18,19] and references cited therein. It is also interesting to observe a close relation between the QE constants and the distance spectra. In fact, for a finite connected graph we have where λ 1 (G) and λ 2 (G) are respectively the largest and the second largest eigenvalues of the distance matrix of G. It is straightforward to see that λ 2 (G) = QEC(G) holds if the distance matrix of G has a constant row sum (in some literatures, such a graph is called transmission regular). But the converse is not true as the paths P n with even n provide counter-examples. In this aspect characterization of graphs satisfying λ 2 (G) = QEC(G) is an interesting problem, as is suggested by the attempt of classifying graphs in terms of the second largest eigenvalue λ 2 (G), see [14].
In this paper, we initiate the project of characterizing finite connected grahs in terms of the QE constants. Our idea is based on the fact that the QE constants of paths form a strictly increasing sequence: QEC(P 2 ) < QEC(P 3 ) < · · · < QEC(P n ) < QEC(P n+1 ) < · · · → − 1 2 . (1.2) Then a natural question arises to determine finite connected graphs along the above QE constants. More precisely, we are interested in the family of graphs G satisfying QEC(P n ) ≤ QEC(G) < QEC(P n+1 ), n ≥ 2. (1. 3) The main goal of this paper is to give the answer to the first two cases of n = 2, 3. This paper is organized as follows: In Section 2 we give a quick review on the QE constant, for more details see [17,21].
In Section 3 we derive a general criterion for the strict inequality where G ⋆ K m+1 is the star product, namely, the graph obtained by joining a graph G and the complete graph K m+1 at a single vertex, see Theorem 3.1. We then prove that the QE constants of paths form a strictly increasing sequence as in (1.2), see Theorem 3.4. In Section 4 we prove the main results. Case of n = 2 is simple, in fact, condition (1.3) characterizes the complete graphs, see Theorem 4.6. For a general case the first useful result is that any graph with QEC(G) < −1/2 is diamond-free, claw-free, C 4 -free and C 5 -free, see Corollary 4.4. Then, using the explicit values of QEC(K m ⋆ K n ) we obtain an explicit list for case of n = 3, that is, a series of graphs K n ⋆ K 2 with n ≥ 2 and one sporadic K 3 ⋆ K 3 , see Theorem 4.11. As a result, QEC(P 4 ) is the smallest accumulation point of the QE constants. We also provide examples of graphs G satisfying QEC(G) = QEC(P 4 ).
Acknowledgements: NO thanks Institut Teknologi Bandung for their kind hospitality, where this work was completed in March 2019. The support by JSPS Open Partnership Joint Research Project "Extremal graph theory, algebraic graph theory and mathematical approach to network science" (2017-18) is gratefully acknowledged. He also thanks Professor J. Koolen for stimulating discussion.

Definition and Basic Properties
A graph G = (V, E) is a pair of a non-empty set V of vertices and a set E of edges, i.e., E is a subset of {{i, j} ; i, j ∈ V, i j}. A graph is called finite if V is a finite set. Throughout this paper by a graph we mean a finite graph.
If {i, j} ∈ E, we write i ∼ j for simplicity. A finite sequence of vertices i 0 , i 1 , . . . , i m ∈ V is called an m-step walk if i 0 ∼ i 1 ∼ · · · ∼ i m . In that case we say that i 0 and i m are connected by a walk of length m. A graph is called connected if any pair of vertices are connected by a walk.
Let G = (V, E) be a connected graph. For i, j ∈ V with i j let d(i, j) = d G (i, j) denote the length of a shortest walk connecting i and j. By definition we set d(i, i) = 0. Then d(i, j) becomes a metric on V, which we call the graph distance. The diameter of G is defined by The distance matrix of G is defined by Let G = (V, E) be a connected graph with |V| ≥ 2. The quadratic embedding constant (QE constant for short) of G is defined by where C(V) is the space of all R-valued functions on V and ·, · the canonical inner product on C(V). Furthermore, 1 is the constant function defined by 1(x) = 1 for all x ∈ V, and 1, f = x∈V f (x). Indeed, identifying C(V) with R n , n = |V|, we see that the domain is a compact manifold (in fact, a sphere of n − 2 dimension). Hence the quadratic function f, D f attains the maximum on the above domain.
(ii) D is conditionally negative definite, that is, The map ϕ : V → H in the above condition (i) is called a quadratic embedding of G. The above result is essentially due to Schoenberg [22,23] and motivated us to introduce the QE constant.
The graphs of QE class include the complete graphs K n (n ≥ 2), paths P n (n ≥ 2), and cycles C n (n ≥ 3). In fact, and QEC(C 2n+1 ) = − 1 4 cos 2 π 2n + 1 , while a closed expression for QEC(P n ) is not known. It is also noted that the QE constant of a tree is negative. In fact, for any tree G on n vertices we have However, (2.4) is a rather rough estimate and its refinement is an interesting question, see [17,Section 5]. Proof. Take f ∈ C(W) such that where ·, · W denotes the inner product on C(W).
The proofs are straightforward from Proposition 2.2. In fact, as is shown in Subsection 3.2, the inequalities in (2.5) are strict.
Next we derive a useful criterion for isometric embedding. (1) If H is isometrically embedded, then H is an induced subgraph of G.

(2) If H is an induced subgraph of G and diam (H) ≤ 2, then H is isometrically embedded in G.
Proof. Let d G and d H be the graph distances of G and H, respectively.
(1) Let i, j ∈ W and assume that they are adjacent in G. Then d G (i, j) = 1 and by assumption we have d H (i, j) = 1, which means that i and j are adjacent in H too. Therefore, H is an induced subgraph of G.
Proposition 2.6. Let G be a connected graph, and H a connected and induced subgraph of G. If diam(H) ≤ 2, we have Proof. It follows from Lemma 2.5 (2) that H is isometrically embedded in G. Then, by Proposition 2.2 we see that QEC(H) ≤ QEC(G).

Calculating QE Constants
Let G be a connected graph on V = {1, 2, . . . , n} and identify C(V) with R n in a natural manner. Recall that QEC(G) is the conditional maximum of the quadratic The method of Lagrange multipliers is applied to calculating QE constants. For later use we review it quickly, for more details see [21]. First we set Since conditions (2.6) and (2.7) define a sphere of n − 2 dimension, which is smooth and compact, the conditional maximum of f, D f under question is attained at a stationary points of F( f, λ, µ). Let S be the set of stationary points of F( f, λ, µ), that is, Taking the derivatives of (2.8), we obtain Thus, S is the set of ( f, λ, µ) ∈ R n × R × R satisfying (2.6), (2.7) and (2.9). On the other hand, for ( f, λ, µ) ∈ S we have Thus we come to the following useful result.

QE Constants of Paths
Let G 1 and G 2 be two graphs with disjoint vertex sets. Choose o 1 and o 2 as distinguished vertices of G 1 and G 2 , respectively. A star product of G 1 and G 2 with respect to o 1 and o 2 is (informally) defined to be the graph obtained by joining G 1 and G 2 at the distinguished vertices o 1 and o 2 . If there is no danger of confusion, the star product is denoted simply by G 1 ⋆ G 2 .
In this subsection we consider the case where G 1 is an arbitrary connected graph and G 2 a complete graph. To be precise, for n ≥ 2 and m ≥ 1 let G = (V, E) be a connected graph on V = {1, 2, . . . , n} and K m+1 the complete graph on {n, n + 1, . . . , n + m}. We set ThenG = (Ṽ,Ẽ) becomes the star product of G and K m+1 , which we denote simply byG = G ⋆ K m+1 . Since G is isometrically embedded inG, it follows from Proposition 2.2 that We are interested in when the inequality (3.1) becomes strict.

3)
J the matrix whose entries are all one and I the identity matrix.
Proposition 3.2. Let G 1 and G 2 be connected graphs with QEC(G 1 ) < 0 and QEC(G 2 ) < 0. Then For the proof see [17,Section 4], where a more precise estimate is obtained.
Proof of the right-half of (3.6). Note that QEC(K m+1 ) = −1 for all m ≥ 1. It then follows immediately from Proposition 3.2 that Here condition (3.5) is not necessary.

Remark 3.3.
For the strict inequality of the left-half of (3.6) condition (3.5) is necessary. We give a simple example. Consider the graph G on five verices and G = G ⋆ K 2 on six vertices as is illustrated in Figure 2. By direct computation we easily obtain

In fact, QEC(G) is attained by
.
In this section we prove that the above inequality is strict.

Proof. The distance matrix of P n is given by
According to the general method described in Subsection 2.2 let S be the set of ( f, λ, µ) ∈ R n × R × R such that (3.23) Then λ 0 = QEC(P n ) is the maximum of λ ∈ R such that ( f, λ, µ) ∈ S for some f ∈ R n and µ ∈ R. It is readily known that λ 0 < 0. By virtue of Theorem 3.1 it is sufficient to show that there exists ( f 0 = [ f 0 (i)], λ 0 , µ 0 ) ∈ S such that f 0 (n) 0. In fact, we will prove a slightly stronger result: for (3.24) where 2 ≤ k ≤ n. We will derive f (k − 1) = 0. The k-th coordinate of (3.21) is given by and by assumption (3.24) we have Similarly, looking at the (k − 1)-th coordinate of (3.21), we obtain On the other hand, by (3.23) and (3.24) we have (3.28) Comparing (3.26) and (3.28), we obtain Since λ ≤ λ 0 < 0, we obtain f (k −1) = f (k) = 0 as desired. Thus, by induction we see that f (n) = 0 implies that f ( j) = 0 for all 1 ≤ j ≤ n, which is in contradiction to condition (3.22). Consequently, f (n) 0 for any ( f, λ, µ) ∈ S. For the proof see [17,Section 5], where a precise estimate of QEC(P n ) from below is obtained.
Our main interest along (4.1) is to characterize the family of graphs G satisfying in terms of geometric or combinatorial properties of graphs. We are also interested in the graphs G satisfying We first recall the following simple fact mentioned in Corollary 2.4 (2). Next we provide simple criteria for (4.3) in terms of forbidden subgraphs. Let K 4 \{e} denote the diamond, that is, the graph obtained by deleting one edge from the complete graph K 4 , see Figure 3. Let K m,n denote the complete bipartite graph with two parts of m and n vertices. In particular, K 1,n is called a star and K 1,3 a claw, see Figure 3.  The following result is immediate from Propositions 4.2 and 4.3.
Remark 4.5. As an immediate consequence from Corollary 4.4, the family of graphs with QEC(G) < −1/2 forms a subfamily of the claw-free graphs. On the other hand, claw-free graphs has been actively studied with various classifications, see e.g., [8]. It would be interesting to revisit the classification of claw-free graphs along with QEC(P n ).
Proof. Suppose that a graph G satisfies (4.4). Then by Proposition 4.1, we have diam(G) = 1, which means that G is a complete graph. On the other hand, it is known that QEC(K n ) = −1 = QEC(P 2 ) for all n ≥ 2. The assertion is then obvious.

Calculating QEC(K n ⋆ K m )
We consider the star product of two complete graphs K n and K m , see Figure 4. To be precise, let n ≥ 1 and m ≥ 2, and consider the graphsG = (Ṽ,Ẽ), wherẽ Obviously, we haveG = K n ⋆ K m , where the induced subgraphs spanned by {1, 2, . . . , n} and by {n, n + 1, . . . , n + m − 1} are the complete graphs K n and K m , respectively. LetD be the distance matrix ofG = K n ⋆ K m . It is convenient to writeD in the block matrices: subject to f ,f = 1, According to the block diagonal expression (4.5), we writef = [ f g] T , where f ∈ R n , g ∈ R m−1 . Then (4.6) becomes where we used Define and let S be the set of its stationary points ( f, g, λ, µ) ∈ R n × R m−1 × R × R, that is the solutions to Keeping in mind that −1 < QEC(G) < 0 unless m = 1 or n = 1, we find after simple calculus that the maximum of λ appearing in the solution is which coincides with QEC(G) by the general theory mentioned in Subsection 2.2.
We have thus obtained the following result.
Proof. The inequality QEC(K n ⋆ K m ) < QEC(P 4 ) is equivalent to of which integer solutions are obtained easily by simple algebra. (ii) m ≥ 2 and n = 2; (iii) m = n = 3.
The equality in (4.9) occurs only when m = n = 2.

Determining the class QEC(P 3 ) ≤ QEC(G) < QEC(P 4 )
This subsection is devoted to the proof of the following result. if and only if G is a star product K n ⋆ K 2 with n ≥ 2 or K 3 ⋆ K 3 . Moreover, In particular, QEC(G) = QEC(P 3 ) if and only if G = P 3 = K 2 ⋆ K 2 . Proof. It follows from Corollary 2.4 that diam(G) ≤ 2. If diam(G) = 1, then G is a complete graph and QEC(G) = −1, which does not satisfy (4.10). Hence, necessarily diam(G) = 2 and |V| ≥ 3.
In general, a clique of G is an induced subgraph of G which is isomorphic to a complete graph. A clique K = (W, F) is called maximal if there is no clique containing K properly. A maximal clique K = (W, F) is called largest or maximum if there is no clique on |W|+1 vertices. Clearly, any graph contains a largest clique. Proof. Since G is not a complete graph by Lemma 4.12, we have W V. That |W| ≥ 2 follows from |V| ≥ 3.
Lemma 4.14. Let G = (V, E) be a connected graph with |V| ≥ 2 and QEC(G) < −1/2, and K = (W, F) a maximal clique. Then for any pair a ∈ V\W and a ′ ∈ W with a ∼ a ′ we have {x ∈ W ; x ∼ a} = {a ′ }.
Proof. (Note that the assertion is trivial if W = V.) Given a pair a ∈ V\W and a ′ ∈ W with a ∼ a ′ , we set s = |{x ∈ W ; x ∼ a}|. Obviously, 1 ≤ s < |W|. We will show by contradiction that s = 1. Suppose that s ≥ 2. Then there exist three distinct vertices x 1 , x 2 , y ∈ W such that a ∼ x 1 , a ∼ x 2 and a y. Note that the induced subgraph spanned by {a, x 1 , x 2 , y} is isomorphic to a diamond K 4 \{e}. It then follows immediately from Proposition 4.2 that QEC(G) ≥ −1/2, which is in contradiction to the assumption QEC(G) < −1/2. Proof. If a = b the assertion follows immediately from Lemma 4.14. We consider the case of a b. To prove the assertion by contradiction, we assume that a ′ b ′ . Since d(a, b) ≤ diam(G) = 2, we have two cases: d(a, b) = 1 or d(a, b) Suppose first that d(a, b) = 1, that is, a ∼ b. Then the induced subgraph spanned by {a, a ′ , b ′ , b} is isomorphic to C 4 , which is a forbidded subgraph by Corollary 4.4. Hence d(a, b) = 1 does not happen. Suppose next that d(a, b) = 2. Then there exists c ∈ V such that a ∼ c ∼ b. Since a b ′ and b a ′ by Lemma 4.14, we have c a ′ , b ′ and c W. There are four cases: (i) c a ′ and c b ′ . The induced subgraph spanned by {a, a ′ , b ′ , b, c} is isomorphic to C 5 , which is a forbidded subgraph by Corollary 4.4.
(ii) c a ′ and c ∼ b ′ . The induced subgraph spanned by {a, a ′ , b ′ , c} is isomorphic to C 4 , which is a forbidded subgraph by Corollary 4.4.
(iii) c ∼ a ′ and c b ′ . This case is similar to (ii). (iv) c ∼ a ′ and c ∼ b ′ . This does not happen by virtue of Lemma 4.14.
In any case we come to contradiction and the proof is completed.
Proof of Theorem 4.11. Let G = (V, E) be a connected graph satisfying (4.10) K = (W, F) be a largest clique with m = |W|. Note that V W and m ≥ 2 by Lemma 4.13. Now divide V\W into two subsets: where U 1 is the set of vertices a ∈ V\W which are directly connected to vertices in W, and U 2 the rest, see Figure 5. Obviously, U 1 ∅. Moreover, by Lemma 4.15 there exists a unique a ′ ∈ W such that a ∼ a ′ for all a ∈ U 1 .
We first prove that U 2 = ∅. Suppose otherwise. Take x ∈ W with x a ′ and y ∈ U 2 . Then we have d(x, y) ≥ 3, which is in contradiction to diam(G) = 2.
We next prove that any pair of vertices a, b ∈ U 1 , a b, are connected by an edge. Suppose otherwise. Take x ∈ W with x a ′ and consider the induced subgraph spanned by {x, a ′ , a, b} is isomorphic to K 1,3 , which is a forbidded subgraph by Corollary 4.4.
Consequently, The induced subgraph spanned by U 1 is a complete graph on |U 1 | ≥ 1 vertices. Hence G is necessarily a star product of two complete graphs: G = K m ⋆ K |U 1 |+1 . Then the assertion follows from Corollary 4.10.
The induced subgraph spanned by {1, 2, . . . , n} is the complete graph K n . We write G = BK n,m and call it a bearded complete graph. The distance matrix D of G = BK n,m is written in the block matrices: For m = 1 and n ≥ 2 we have BK n,1 = K n ⋆ K 2 = K n ∧ K 1,1 . It is already known that QEC(BK n,1 ) = − 2 2 + 2 1 − 1 n .
The above formula is valid for n = 1.