20 research outputs found
On unicyclic graphs whose second largest eigenvalue dose not exceed 1
AbstractConnected graphs in which the number of edges equals the number of vertices are called unicyclic graphs. In this paper, all unicyclic graphs whose second largest eigenvalue does not exceed 1 have been determined
Laplacian spectral characterization of some double starlike trees
A tree is called double starlike if it has exactly two vertices of degree
greater than two. Let denote the double starlike tree obtained by
attaching pendant vertices to one pendant vertex of the path and
pendant vertices to the other pendant vertex of . In this paper, we prove
that is determined by its Laplacian spectrum
Spectral characterizations of sun graphs and broken sun graphs
Graphs and AlgorithmsInternational audienceSeveral matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns
The inertia of unicyclic graphs and the implications for closed-shells
AbstractThe inertia of a graph is an integer triple specifying the number of negative, zero, and positive eigenvalues of the adjacency matrix of the graph. A unicyclic graph is a simple connected graph with an equal number of vertices and edges. This paper characterizes the inertia of a unicyclic graph in terms of maximum matchings and gives a linear-time algorithm for computing it. Chemists are interested in whether the molecular graph of an unsaturated hydrocarbon is (properly) closed-shell, having exactly half of its eigenvalues greater than zero, because this designates a stable electron configuration. The inertia determines whether a graph is closed-shell, and hence the reported result gives a linear-time algorithm for determining this for unicyclic graphs
The inertia of weighted unicyclic graphs
Let be a weighted graph. The \textit{inertia} of is the triple
, where
are the number of the positive, negative and zero
eigenvalues of the adjacency matrix of including their
multiplicities, respectively. , is called the
\textit{positive, negative index of inertia} of , respectively. In this
paper we present a lower bound for the positive, negative index of weighted
unicyclic graphs of order with fixed girth and characterize all weighted
unicyclic graphs attaining this lower bound. Moreover, we characterize the
weighted unicyclic graphs of order with two positive, two negative and at
least zero eigenvalues, respectively.Comment: 23 pages, 8figure
On the spectral moments of unicyclic graphs with fixed diameter
AbstractUnicyclic graphs are connected graphs in which the number of edges equals the number of vertices. Let Un,d be the class of unicyclic graphs of order n and diameter d. For unicyclic graphs, lexicographic ordering by spectral moments (S-order) is discussed in this paper. The last d+⌊d2⌋-2 graphs, in the S-order, among all unicyclic graphs in Un,d(3⩽d⩽n-5) are characterized. In addition, for the cases d=n-4 and d=n-3, the last 2n-11 and last n+⌊n2⌋-7 unicyclic graphs in the set Un,d are also given, respectively