Xiang-Feng Pan

Path embedding in faulty hypercubes

Abstract There are some interesting results concerning longest paths or even cycles embedding in faulty hypercubes. This paper considers the embeddings of paths of all possible lengths between any two fault-free vertices in faulty hypercubes. Let fv (respectively, fe) denote the number of faulty vertices (respectively, edges) in an n-dimensional hypercube Qn. We prove that there exists a fault-free path of length l between any two distinct fault-free vertices u and v in Qn with f v + f e ⩽ n - 2 for each l satisfying d Q n ( u , v ) + 2 ⩽ l ⩽ 2 n - 2 f v - 1 and 2 | ( l - d Q n ( u , v ) ) . The bounds on path length l and faulty set size f v + f e for a successful embedding are tight. That is, the result does not hold if l d Q n ( u , v ) + 2 or l > 2 n - 2 f v - 1 or f v + f e > n - 2 . Moreover, our result improves some known results.

Asymptotic Laplacian-energy-like invariant of lattices

Let µ 1 ? µ 2 ? ? ? µ n denote the Laplacian eigenvalues of a graph G with n vertices. The Laplacian-energy-like invariant, denoted by LEL ( G ) = ? i = 1 n - 1 µ i , is a novel topological index. In this paper, we show that the Laplacian-energy-like per vertex of various lattices is independent of the toroidal, cylindrical, and free boundary conditions. Simultaneously, the explicit asymptotic values of the Laplacian-energy-like in these lattices are obtained. Moreover, our approach implies that in general the Laplacian-energy-like per vertex of other lattices is independent of the boundary conditions.

Complete characterization of bicyclic graphs with minimal Kirchhoff index

The resistance distance between any two vertices of a graph G is defined as the network effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index K ( G ) is the sum of the resistance distances between all the pairs of vertices in G . A bicyclic graph is a connected graph whose number of edges is exactly one more than its number of vertices. In this paper, we completely characterize the bicyclic graphs of order n ? 4 with minimal Kirchhoff index and determine bounds on the Kirchhoff index of bicyclic graphs. This improves and extends some earlier results.

On degree resistance distance of cacti

A graph G is called a cactus if each block of G is either an edge or a cycle. Denote by C a c t ( n ; t ) the set of connected cacti possessing n vertices and t cycles. In a recent paper (Du et?al., 2015), the C a c t ( n ; t ) with minimum degree resistance distance was characterized. We now determine the elements of C a c t ( n ; t ) with second-minimum and third-minimum degree resistance distances. In addition, some mistakes in Du et?al. (2015) are pointed out.

Minimizing Kirchhoff index among graphs with a given vertex bipartiteness

The resistance distance between any two vertices of a graph G is defined as the effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index Kf(G) is the sum of the resistance distances between all the pairs of vertices in G. The vertex bipartiteness vb of a graph G is the minimum number of vertices whose deletion from G results in a bipartite graph. In this paper, we characterize the graph having the minimum Kf(G) values among graphs with a fixed number n of vertices and fixed vertex bipartiteness, 1 ź v b ź n - 3 .

A unified approach to the asymptotic topological indices of various lattices

We present a unified approach to the asymptotic topological indices of various lattices.We propose the topological indices per vertex problem for lattice systems.The explicit asymptotic values of Laplacian energies for various lattices are obtained.We deduce the Laplacian energies of many types of lattices are independent of various boundary conditions. In this paper, we present a unified approach to the asymptotic topological indices of various lattices. Moreover, we propose the various topological indices per vertex problem for lattice systems and show that the various topological indices per vertex of lattices are independent of the toroidal, cylindrical, and free boundary conditions. Our result is a generalization of some earlier results.

The spectral moments of trees with given maximum degree

Abstract Let Δ ≥ 3 . Denote by T n , Δ the set of all trees with n vertices and maximum degree Δ and by T n , Δ ∗ the set of all Δ -trees with n vertices. In this work, we first show that all Δ -trees come before all trees in T n , Δ ∖ T n , Δ ∗ in an S -order and present a criterion for a Δ -tree coming before another Δ -tree in an S -order. Also, we give the first ∑ k = 1 ⌊ n − 1 3 ⌋ ( ⌊ n − k − 1 2 ⌋ − k + 1 ) graphs apart from a path, in an S -order, of all trees with n vertices.

The {1}-inverse of the Laplacian of subdivision-vertex and subdivision-edge coronae with applications

The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let and be two vertex disjoint graphs. The subdivision-vertex corona of and , denoted by , is the graph obtained from and copies of , all vertex-disjoint, by joining the ith vertex of to every vertex in the ith copy of . The subdivision-edge corona of and , denoted by , is the graph obtained from and copies of , all vertex-disjoint, by joining the ith vertex of to every vertex in the ith copy of , where is the set of inserted vertices of . In this paper, -inverse for the Laplacian matrices of graphs and are proposed, based on which the explicit resistance distance can be obtained for the two-vertex resistance between arbitrary vertices in the graphs.

Sharp bounds of the zeroth-order general Randić index of bicyclic graphs with given pendent vertices

Let G be a simple connected graph and α be a given real number. The zeroth-order general Randic index of 0 R α ( G ) is defined as ? v ? V ( G ) d G ( v ) ] α , where d G ( v ) denotes the degree of the vertex v of G . In this paper, for any α ( ? 0 , 1 ) , we give sharp bounds of the zeroth-order general Randic index 0 R α of all bicyclic graphs with n vertices and k pendent vertices.

Sharp bounds on the zeroth-order general Randić index of unicyclic graphs with given diameter

Abstract Let G be a simple connected graph and α be a given real number. The zeroth-order general Randic index 0 R α ( G ) is defined as ∑ v ∈ V ( G ) [ d G ( v ) ] α , where d G ( v ) denotes the degree of the vertex v of G . In this work, we give, for any α ( ≠ 0 , 1 ) , some sharp bounds on the zeroth-order general Randic index 0 R α of all unicyclic graphs with n vertices and diameter d .