Jianxi Li

The harmonic index of a graph

The harmonic index of a graph G is defined as the sum of weights 2 d(vi)+d(vj) of all edges vivj of G, where d(vi) denotes the degree of the vertex vi in G. In this paper, we study how the harmonic index behaves when the graph is under perturbations. These results are used to provide a simpler method for determining the unicyclic graphs with maximum and minimum harmonic index among all unicyclic graphs, respectively. Moreover, a lower bound for harmonic index is also obtained.

An edge-separating theorem on the second smallest normalized Laplacian eigenvalue of a graph and its applications

Abstract Let λ 2 ( G ) be the second smallest normalized Laplacian eigenvalue of a graph G . In this paper, we investigate the behavior on λ 2 ( G ) when the graph G is perturbed by separating an edge. This result can be used to determine all trees and unicyclic graphs with λ 2 ( G ) ≥ 1 − 2 2 . Moreover, the trees and unicyclic graphs with λ 2 ( G ) = 1 − 2 2 are also determined, respectively.

The number of spanning trees of a graph

In this paper, we present some sharp upper bounds for the number of spanning trees of a connected graph in terms of its structural parameters such as the number of vertices, the number of edges, maximum vertex degree, minimum vertex degree, connectivity and chromatic number.

The smallest values of algebraic connectivity for unicyclic graphs

The algebraic connectivity of G is the second smallest eigenvalue of its Laplacian matrix. Let Un be the set of all unicyclic graphs of order n. In this paper, we will provide the ordering of unicyclic graphs in Un up to the last seven graphs according to their algebraic connectivities when n>=13. This extends the results of Liu and Liu [Y. Liu, Y. Liu, The ordering of unicyclic graphs with the smallest algebraic connectivity, Discrete Math. 309 (2009) 4315-4325] and Guo [J.-M. Guo, A conjecture on the algebraic connectivity of connected graphs with fixed girth, Discrete Math. 308 (2008) 5702-5711].

Bounds on normalized Laplacian eigenvalues of graphs

Let G be a simple connected graph of order n, where n≥2. Its normalized Laplacian eigenvalues are 0=λ1≤λ2≤⋯≤λn≤2. In this paper, some new upper and lower bounds on λn are obtained, respectively. Moreover, connected graphs with λ2=1 (or λn−1=1) are also characterized.MSC:05C50, 15A48.

Effects on the normalized Laplacian spectral radius of non-bipartite graphs under perturbation and their applications

The normalized Laplacian eigenvalues of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As an example of the application, the smallest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order is determined.

THE MINIMUM ALGEBRAIC CONNECTIVITY OF CATERPILLAR UNICYCLIC GRAPHS

A caterpillar unicyclic graph is a unicyclic graph in which the removal of all pendant vertices makes it a cycle. In this paper, the unique caterpillar unicyclic graph with minimum algebraic connectivity among all caterpillar unicyclic graphs is determined.

Bounding the sum of powers of normalized Laplacian eigenvalues of a graph

Let G be a simple connected graph of order n. Its normalized Laplacian eigenvalues are λ1≥λ2≥⋯≥λn−1≥λn=0. In this paper, new bounds on Sβ*(G)=∑i=1n−1λiβ (β ≠ 0, 1) are derived.

Coefficients of the Characteristic Polynomial of the (Signless, Normalized) Laplacian of a Graph

In this paper, we give a combinatorial expression for the fifth coefficient of the (signless) Laplacian characteristic polynomial of a graph. The first five normalized Laplacian coefficients are also given.

The spectral radii of graphs with prescribed degree sequence

In this paper, we first present the properties of the graph which maximize the spectral radius among all graphs with prescribed degree sequence. Using these results, we provide a somewhat simpler method to determine the unicyclic graph with maximum spectral radius among all unicyclic graphs with a given degree sequence. Moreover, we determine the bicyclic graph which has maximum spectral radius among all bicyclic graphs with a given degree sequence. Let G be a simple connected graph with vertex set V (G) and edge set E(G). Its order is |V (G)|, denoted by n, and its size is |E(G)|, denoted by m. For v ∈ V (G), let NG(v) (or N (v) for short) be the set of all neighbors of v in G and let d(v) = |N (v)| be the degree of v. We use G − e and G + e to denote the graphs obtained by deleting the edge e from G and by adding the edge e to G, respectively. For any nonempty subset W of V (G), the subgraph of G induced by W is denoted by G(W ). The distance of u and v (in G) is the length of the shortest path between u and v, denoted by d(u;v). For all other notions and definitions, not given here, see, for example, (1), or (4) (for graph spectra). For the basic notions and terminology on the spectral graph theory the readers are referred to (4). Let A(G) be the adjacency matrix of G. Its eigenvalues are called the eigenvalues