27 research outputs found
Laplacian spectral properties of signed circular caterpillars
A circular caterpillar of girth n is a graph such that the removal of all pendant vertices yields a cycle Cn of order n. A signed graph is a pair Γ = (G, σ), where G is a simple graph and σ ∶ E(G) → {+1, −1} is the sign function defined on the set E(G) of edges of G. The signed graph Γ is said to be balanced if the number of negatively signed edges in each cycle is even, and it is said to be unbalanced otherwise. We determine some bounds for the first n Laplacian eigenvalues of any signed circular caterpillar. As an application, we prove that each signed spiked triangle (G(3; p, q, r), σ), i. e. a signed circular caterpillar of girth 3 and degree sequence πp,q,r = (p + 2, q + 2, r + 2, 1,..., 1), is determined by its Laplacian spectrum up to switching isomorphism. Moreover, in the set of signed spiked triangles of order N, we identify the extremal graphs with respect to the Laplacian spectral radius and the first two Zagreb indices. It turns out that the unbalanced spiked triangle with degree sequence πN−3,0,0 and the balanced spike triangle (G(3; p, ^ q, ^ r^), +), where each pair in {p, ^ q, ^ r^} differs at most by 1, respectively maximizes and minimizes the Laplacian spectral radius and both the Zagreb indices
A lower bound for the first Zagreb index and its application
For a graph G, the first Zagreb index is defined as the sum of the squares of the vertices degrees. By investigating the connection between the first Zagreb index and the first three coefficients of the Laplacian characteristic polynomial, we give a lower bound for the first Zagreb index, and we determine all corresponding extremal graphs. By doing so, we generalize some known results, and, as an application, we use these results to study the Laplacian spectral determination of graphs with small first Zagreb index
Signed bicyclic graphs with minimal index
The index of a signed graph \Sigma = (G; \sigma) is just the largest eigenvalue
of its adjacency matrix. For any n > 4 we identify the signed graphs achieving the
minimum index in the class of signed bicyclic graphs with n vertices. Apart from the n = 4 case, such graphs are obtained by considering a starlike tree with four branches of suitable length (i.e. four distinct paths joined at their end vertex u) with two additional negative independent edges pairwise joining the four vertices adjacent to u. As a by-product, all signed bicyclic graphs containing a theta-graph and whose index is less than 2 are detected
Ordering signed graphs with large index
The index of a signed graph is the largest eigenvalue of its adjacency matrix. We establish the first few signed graphs ordered decreasingly by the index in classes of connected signed graphs, connected unbalanced signed graphs and complete signed graphs with a fixed number of vertices
Topological and spectral properties of random digraphs
We investigate some topological and spectral properties of
Erd\H{o}s-R\'{e}nyi (ER) random digraphs . In terms of topological
properties, our primary focus lies in analyzing the number of non-isolated
vertices as well as two vertex-degree-based topological indices: the
Randi\'c index and sum-connectivity index . First, by
performing a scaling analysis we show that the average degree serves as scaling parameter for the average values of ,
and . Then, we also state expressions relating the number of arcs,
spectral radius, and closed walks of length 2 to , the parameters of ER
random digraphs. Concerning spectral properties, we compute six different graph
energies on . We start by validating as the scaling
parameter of the graph energies. Additionally, we reformulate a set of bounds
previously reported in the literature for these energies as a function .
Finally, we phenomenologically state relations between energies that allow us
to extend previously known bounds
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure