Weighted Graphs: Eigenvalues and Chromatic Number

We revisit Hoffman relation involving chromatic number $\chi$ and eigenvalues. We construct some graphs and weighted graphs such that the largest and smallest eigenvalues $\lambda$ dan $\mu$ satisfy $\lambda=(1-\chi)\mu.$ We study in particular the eigenvalues of the integer simplex $T_m^2,$ a 3-chromatic graph on $\binom {m+2}{2}$ vertices.


Chromatic number and eigenvalues
Proposition 1.1. (a variant of Hoffman's [4]) Let G be a connected graph, whose edges may have positive weights, endowed with a proper coloration using χ colors. Let G ij be the bipartite graph induced in G by the two color classes i and j. Let K 1 ≥ K 2 ≥ · · · ≥ K χ(χ−1)/2 be the largest eigenvalues of the χ(χ − 1)/2 bipartite graphs G ij . The largest adjacency eigenvalue λ of G is at most the sum χ−1 i=1 K i of the χ − 1 largest eigenvalues of the bipartite graphs. As a corollary, λ ≤ −(χ − 1)µ, where µ is the smallest eigenvalue of G. Moreover, if λ = −(χ − 1)µ, all bipartite graphs defined by two colors have extremal eigenvalues µ and −µ.

General constructions
Obviously, multiplication by a positive real of all weights preserve the property. We observe that if we cut a symmetric matrix into blocks, say with M 11 and M 22 square, of sizes n 1 and n 2 , the matrix of size n 1 + nn 2 has spectrum the multiset union of the spectrum of M and n − 1 copies of the spectrum of M 22 . Thus replication of parts of a (weighted) graph with λ = (1 − χµ) and appropriate modifications of weights give new weighted graphs with the same property. Using these two methods we obtain from the unweighted K 3 some 3-colorable weighted graphs with λ = −2µ ( Figure 2). The unmarqued edges have weight 1.
The graph of order 4 has λ = 2 √ 2; the ones of order 5 have (from left to right) λ = 2 √ 2, λ = 2 √ 3, λ = 4, and λ = 2 √ 3. However the last graphs is not obtained by the same method. We now give some other graphs where Hoffman's inequality becomes an equality. If G has chromatic number χ and satisfies λ G = (1 − χ)µ G ; and if H is a connected graph with at least a loop and two vertices, such that µ H (χ − 1) ≤ λ H , then the categorical product G × H (called product in ([1], www.ejgta.org  p. 66) has also chromatic number χ and satisfies λ G×H = (1 − χ)µ G×H . Morever, with the same condition on H, if G is uniquely χ-colorable, then G × H is also uniquely χ-colorable.
As an example, we show the product with G a triangle and H a path of length 2 with a loop at  The case of G and H complete graphs, with a loop at each vertex of H gives again the regular multipartite complete graphs. Some Cayley graphs have the property. For example the ones built on the groups Z/3pZ × Z/3pqZ with generator {±(1, 0), ±(0, 1), ±(1, 1)} and parameters λ = 6, µ = −3, and χ = 3.
In the same vein, we have a graph T 2 3 (that will appear in Section 3) decomposable into C 6 and two copies ofẼ 6 . The cartesian product of two graphs of chromatic number χ has also chromatic number χ. The condition λ = (1− χ)µ is also preserved by cartesian product, as well as connexity, but the unicity of the χ-coloration is not (unless χ ≤ 2).

Integer simplexes of dimension 2
Here we describe a family of non-regular graphs with the property λ = −2µ, the integer simplexes of dimension 2, denoted T 2 m , that are not regular if m ≥ 2 (see [8]).

Relations
We will prove that the eigenvalues of these graphs satisfy not only the inequalities given in Section 1, but the equality λ = −2µ, and that µ, −µ are the largest and smallest eigenvalues of the three bipartite graphs, even if m is multiple of 3, where the class of color 0 gives a bipartite graph with order 2 + m+2 2 /3 and two (isomorphic) ones of order −1 + m+2 2 /3. Proposition 3.1. The largest adjacency eigenvalue of T 2 m is λ = 2 + 4 cos( 2π m+3 ), and −λ/2 = −1 − 2 cos( 2π m+3 ) is an eigenvalue of T 2 m . Proof. We will consider the polynomials U i defined by U 0 = 1, U 1 = X and U n+2 = XU n+1 −U n . They satisfy the equalities of rational fractions (X −1/X)U n (X +1/X) = X n+1 −X −n−1 and thus the equalities U n (2 cos θ) sin θ = sin((n + 1)θ). They are of course closely related to Chebyshev polynomials of second kind.

Hence the function f that associates to the point
) constitutes an eigenvector for T 2 m , associated to the eigenvalue 2 + 4 cos 2tπ m + 3 .
At this point we may wonder whether −λ/2 = −1 − 2 cos( 2π m+3 ) is the smallest eigenvalue of T 2 m . We know already is the smallest eigenvalue of the three bipartite graphs obtained by removing the vertices in a color class, • the equality µ = −λ/2 holds for small values of m, indeed Maple checked that from m = 1 to m = 20.
But the example of the antiprisms on 3p-gons, p ≥ 3 shows that one cannot already conclude µ = −λ/2.
It appears that the spectrum of G(3(m+3), m+2) contains 3 times the spectrum of T 2 m . Indeed each eigenvector v of T 2 m can be extended into an eigenvector f (v) of G(3(m + 3), m + 2) with the indicated symmetries and null values on the vertices out of the copies of T 2 m ( Figure 5): shifting one step ψ : (x, y) → (x + 1, y) allows to find 3 linearly independent vectors f (v), f (ψ(v)), f (ψ 2 (v)).
Indeed these polynomials are given in Table 1 (with x = sin( π 3m+9 ) and y = cos( π 3m+9 )). Needless to say, for these graphs, since the coarse inequality λ ≤ (1−χ)µ becomes an equality, the refined inequalities as well as the intermediary inequalities also become equalities.

Remark about integer simplexes of higher dimension
The equality λ 1 + (χ − 1)λ n = 0 does not hold for m = 2 and d > 2. Indeed, the highest eigenvalue of T d 2 is d − 1 + √ d 2 + 1, and (d − 1 + √ d 2 + 1/d) is not an algebraic integer. The coloration of T d m is not unique in general: Figure 6 shows two colorations of T 3 2 that are not equivalent under color permutation and not even equivalent under color permutation and graph isomorphism. The same possibility of different colorations for T d m occurs at least when there are several non-isomorphic abelian groups of cardinality d + 1.

Problem
The underlying graphs of examples of Figure 2 are all uniquely 3-colorable graphs on 4 or 5 vertices and even a graph (of size 6) that is not uniquely 3-colorable. Thus the following question is suggested. Does every uniquely 3-colorable graph admit positive weights such that λ = −(χ − 1)µ?
Note that the graph obtained by adding a pendant edge to a triangle (this graph is not uniquely 3-colorable) does not admit such a choice of weights. Indeed, the characteristic polynomial has constant term xy where x and y are the weights of the pendant edge and the one not adjacent to it, but the eigenvalues of the graph should be λ, −λ/2, −λ/2, and 0, that implies a null constant term.