Adjacency Spectral Radius Research Articles

The extremal spectral radii of $$k$$ k -uniform supertrees

In this paper, we study some extremal problems of three kinds of spectral radii of $$k$$k-uniform hypergraphs (the adjacency spectral radius, the signless Laplacian spectral radius and the incidenc...

On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs

In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph G, we construct a generalized power of G, denoted by Gk,s, which is obtained from G by blowing up each vertex into a s-set and each edge into a (k−2s)-set, where s≤k/2. When s<k/2, Gk,s is always odd-bipartite. We show that Gk,k2 is non-odd-bipartite if and only if G is non-bipartite, and find that Gk,k2 has the same adjacency (respectively, signless Laplacian) spectral radius as G. So the results involving the adjacency or signless Laplacian spectral radius of a simple graph G hold for Gk,k2. In particular, we characterize the unique graph with minimum adjacency or signless Laplacian spectral radius among all non-odd-bipartite hypergraphs Gk,k2 of fixed order, and prove that 2+5 is the smallest limit point of the non-odd-bipartite hypergraphs Gk,k2. In addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors.

The Signless Laplacian or Adjacency Spectral Radius of Bicyclic Graphs with Given Number of Cut Edges

Let $$\fancyscript{B}(n,r)$$B(n,r) be the set of all bicyclic graphs with $$n$$n vertices and $$r$$r cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in $$\fancyscript{B}(n,r)$$B(n,r).

Bounds of the Spectral Radius and the Nordhaus-Gaddum Type of the Graphs

The Laplacian spectra are the eigenvalues of Laplacian matrix L(G) = D(G) − A(G), where D(G) and A(G) are the diagonal matrix of vertex degrees and the adjacency matrix of a graph G, respectively, and the spectral radius of a graph G is the largest eigenvalue of A(G). The spectra of the graph and corresponding eigenvalues are closely linked to the molecular stability and related chemical properties. In quantum chemistry, spectral radius of a graph is the maximum energy level of molecules. Therefore, good upper bounds for the spectral radius are conducive to evaluate the energy of molecules. In this paper, we first give several sharp upper bounds on the adjacency spectral radius in terms of some invariants of graphs, such as the vertex degree, the average 2-degree, and the number of the triangles. Then, we give some numerical examples which indicate that the results are better than the mentioned upper bounds in some sense. Finally, an upper bound of the Nordhaus-Gaddum type is obtained for the sum of Laplacian spectral radius of a connected graph and its complement. Moreover, some examples are applied to illustrate that our result is valuable.

Some inequalities on the skew-spectral radii of oriented graphs

Let G be a simple graph and be an oriented graph obtained from G by assigning a direction to each edge of G. The adjacency matrix of G is and the skew-adjacency matrix of is . The adjacency spectral radius of G and the skew-spectral radius of are defined as the spectral radius of and respectively. In this paper, we firstly establish a relation between and . Also, we give some results on the skew-spectral radii of and its oriented subgraphs. As an application of these results, we obtain a sharp upper bound of the skew-spectral radius of an oriented unicyclic graph. MSC:05C50, 15A18.

On the spectral radius of weighted trees with given number of pendant vertices and a positive weight set

Let 𝒯(n, r; W n−1) be the set of all n-vertex weighted trees with r vertices of degree 2 and fixed positive weight set W n−1, 𝒫(n, γ; W n−1) the set of all n-vertex weighted trees with q pendants and fixed positive weight set W n−1, where W n−1 = {w 1, w 2, … , w n−1} with w 1 ⩾ w 2 ⩾ ··· ⩾ w n−1 > 0. In this article, we first identify the unique weighted tree in 𝒯(n, r; W n−1) with the largest adjacency spectral radius. Then we characterize the unique weighted trees with the largest adjacency spectral radius in 𝒫(n, γ; W n−1).

An edge-grafting theorem on Laplacian spectra of graphs and its application

Let 𝒰(n, d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n, d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].

Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity

In this paper, we characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. We also discuss the minimum signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Consequently we give an upper bound of signless Laplacian or adjacency spectral radius of graphs in terms of connectivity. In addition we confirm a conjecture of Aouchiche and Hansen involving adjacency spectral radius and connectivity.

SOME SHARP UPPER BOUNDS ON THE SPECTRAL RADIUS OF GRAPHS

In this paper, we first give a relation between the adjacency spectral radius and the $Q$-spectral radius of a graph. Then using this result, we further give some new sharp upper bounds on the adjacency spectral radius of a graph in terms of degrees and the average 2-degrees of vertices. Some known results are also obtained.

SOME SHARP UPPER BOUNDS ON THE SPECTRAL RADIUS OF GRAPHS

In this paper, we first give a relation between the adjacency spectral radius and the $Q$-spectral radius of a graph. Then using this result, we further give some new sharp upper bounds on the adjacency spectral radius of a graph in terms of degrees and the average 2-degrees of vertices. Some known results are also obtained.