Paper detail

Equivalent characterizations of the spectra of graphs and applications to measures of distance-regularity

As it is well known, the spectrum $ {\rm sp\,} Γ$ (of the adjacency matrix $A$) of a graph $Γ$, with $d$ distinct eigenvalues other than its spectral radius $λ_0$, usually provides a lot of information about the structure of $G$. Moreover, from ${\rm sp\,}Γ$ we can define the so-called predistance polynomials $p_0,\ldots,p_d\in {\mathbb R}_d[x]$, with ${\rm dgr\,} p_i=i$, $i=0,\ldots,d$, which are orthogonal with respect to the scalar product $\langle f, g\rangle_Γ =\frac{1}{n}{\rm tr\,}(f(A)g(A))$ and normalized in such a way that $\|p_i\|_Γ^2=p_i(λ_0)$. They can be seen as a generalization for any graph of the distance polynomials of a distance-regular graph. Going further, we consider the preintersection numbers $ξ_{ij}^h$ for $i,j,h\in\{0,\ldots,d\}$, which generalize the intersection numbers of a distance-regular graph, and they are the Fourier coefficients of $p_ip_j$ in terms of the basis $\{p_h\}_{0\le h\le d}$. The aim of this paper is to show that, for any graph $Γ$, the information contained in its spectrum, predistance polynomials, and preintersection numbers is equivalent. Also, we give some characterizations of distance-regularity which are based on the above concepts. For instance, we comment upon the so-called spectral excess theorem stating that a connected regular graph $G$ is distance-regular if and only if its spectral excess, which is the value of $p_d$ at $λ_0$, equals the average excess, that is, the mean of the numbers of vertices at extremal distance $d$ from every vertex.