174 research outputs found
On outindependent subgraphs of strongly regular graphs
An outindependent subgraph of a graph G, with respect to an independent vertex subset C¿¿V, is the subgraph GC induced by the vertices in V\¿C. We study the case when G is strongly regular, where the results of de Caen [1998, The spectra of complementary subgraphs in a strongly regular graph. European Journal of Combinatorics, 19 (5), 559–565.], allow us to derive the whole spectrum of GC . Moreover, when C attains the Hoffman–Lovász bound for the independence number, GC is a regular graph (in fact, distance-regular if G is a Moore graph). This article is mainly devoted to study the non-regular case. As a main result, we characterize the structure of GC when C is the neighborhood of either one vertex or one edge.Peer ReviewedPostprint (author's final draft
Number of walks and degree powers in a graph
This letter deals with the relationship between the total number of k-walks in a graph, and the sum of the k-th powers of its
vertex degrees. In particular, it is shown that the sum of all k-walks is upper bounded by the sum of the k-th powers of the degrees
The alternating and adjacency polynomials, and their relation with the spectra and diameters of graphs
Let Γ be a graph on n vertices, adjacency matrix A, and distinct eigenvalues λ > λ_1 > λ_2 > · · · > λ_d. For every k = 0,1, . . . ,d −1, the k-alternating polynomial P_k is defined to be the polynomial of degree k and norm |Peer Reviewe
The geometry of t-cliques in k-walk-regular graphs
A graph is walk-regular if the number of cycles of length rooted at a given vertex is a constant through all the vertices.
For a walk-regular graph with different eigenvalues and spectrally maximum diameter , we study the geometry of its
-cliques, that is, the sets of vertices which are mutually at distance . When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a regular tetrahedron and we compute its parameters.
Moreover, the results are generalized to the case of -walk-regular graphs, a family which includes both walk-regular and distance-regular graphs, and their -cliques or vertices at distance from each other
A new approach to the spectral excess theorem for distance-regular graphs
The Spectral Excess Theorem provides a quasi-spectral characterization for a (regular) graph with different eigenvalues to be distance-regular graph, in terms of the mean
(d-1)-excess of its vertices.\ The original approach, due to Fiol and Garriga in , was obtained in a wide context from a local point of view, so giving a characterization of the so-called pseudo-distance-regularity around a vertex.\ In this paper we
present a new simple method based in a global point of view, and where the mean degree of the distance- graph plays an essential role
A differential approach for bounding the index of graphs under perturbations
This paper presents bounds for the variation of the spectral radius (G) of
a graph G after some perturbations or local vertex/edge modifications of G. The
perturbations considered here are the connection of a new vertex with, say, g vertices
of G, the addition of a pendant edge (the previous case with g = 1) and the addition
of an edge. The method proposed here is based on continuous perturbations and
the study of their differential inequalities associated. Within rather economical
information (namely, the degrees of the vertices involved in the perturbation), the
best possible inequalities are obtained. In addition, the cases when equalities are
attained are characterized. The asymptotic behavior of the bounds obtained is
also discussed.Postprint (published version
On k-Walk-Regular Graphs
Considering a connected graph with diameter , we say that it
is \emph{-walk-regular}, for a given integer , if the number of walks of length between vertices
and only depends on the distance between them, provided that
this distance does not exceed . Thus, for , this definition
coincides with that of walk-regular graph, where the number of
cycles of length rooted at a given vertex is a constant
through all the graph. In the other extreme, for , we get one
of the possible definitions for a graph to be distance-regular. In
this paper we present some algebraic characterizations of
-walk-regularity, which are based on the so-called local spectrum
and predistance polynomials of . Moreover, some results, concerning some parameters of a geometric nature, such as the cosines, and the spectrum of walk-regular graphs are presented
Moments in graphs
Let G be a connected graph with vertex set V and a weight function that assigns
a nonnegative number to each of its vertices. Then, the -moment of G at vertex u
is de ned to be M
G(u) =
P
v2V (v) dist(u; v), where dist( ; ) stands for the distance
function. Adding up all these numbers, we obtain the -moment of G:
This parameter generalizes, or it is closely related to, some well-known graph invari-
ants, such as the Wiener index W(G), when (u) = 1=2 for every u 2 V , and the
degree distance D0(G), obtained when (u) = (u), the degree of vertex u.
In this paper we derive some exact formulas for computing the -moment of a
graph obtained by a general operation called graft product, which can be seen as a
generalization of the hierarchical product, in terms of the corresponding -moments
of its factors. As a consequence, we provide a method for obtaining nonisomorphic
graphs with the same -moment for every (and hence with equal mean distance,
Wiener index, degree distance, etc.). In the case when the factors are trees and/or
cycles, techniques from linear algebra allow us to give formulas for the degree distance
of their product.Postprint (author’s final draft
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