Jinlong Shu

A Sharp Upper Bound of the Spectral Radius of Graphs

Let G be a simple connected graph with n vertices and m edges. Let ?(G)=? be the minimum degree of vertices of G. The spectral radius ?(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we obtain the following sharp upper bound of ?(G): ?(G)???1+(?+1)2+4(2m??n)2.Equality holds if and only if G is either a regular graph or a bidegreed graph in which each vertex is of degree either ? or n?1.

A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph

Abstract Let G be a simple connected graph with n vertices. The largest eigenvalue of the Laplacian matrix of G is denoted by μ(G). Suppose the degree sequence of G is d1⩾d2⩾⋯⩾dn. In this paper, we present a sharp upper bound of μ(G) μ(G)⩽d n + 1 2 + d n − 1 2 2 +∑ i=1 n d i (d i −d n ) , the equality holds if and only if G is a regular bipartite graph.

A sharp upper bound for the spectral radius of the Nordhaus-Gaddum type

Abstract Let G be a simple graph with n vertices and let G c be its complement. Let ρ ( G ) be the spectral radius of adjacency matrix A ( G ) of G. In this paper, a sharp upper bound of the Nordhaus–Gaddum type is obtained: ρ(G)+ρ(G c )⩽ 2− 1 k − 1 k n(n−1) , where k and k are the chromatic numbers of G and G c , respectively. Equality holds if and only if G is a complete graph or an empty graph.

Sharp lower bounds of the least eigenvalue of planar graphs

Let G be a simple graph with n⩾3 vertices and orientable genus g and non-orientable genus h. We define the Euler characteristic χ(G) of a graph G by χ(G)=max{2−2g,2−h}. Let λ(G) be the least eigenvalue of the adjacency matrix A of G. In this paper, we obtain the following lower bounds of λ(G) λ(G)⩾−2(n−χ(G)). In particular, if G is the planar graph, then λ(G)⩾−2n−4 the equality holds if and only if G≅K2,n−2. Further, we have same result of series–parallel graph.

Some graft transformations and its applications on the distance spectral radius of a graph

Abstract Let D ( G ) = ( d i , j ) n × n denote the distance matrix of a connected graph G with order n , where d i j is equal to the distance between v i and v j in G . The largest eigenvalue of D ( G ) is called the distance spectral radius of graph G , denoted by ϱ ( G ) . In this paper, some graft transformations that decrease or increase ϱ ( G ) are given. With them, for the graphs with both order n and k pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2 ≤ k ≤ n − 2 ; for k = 1 , 2 , 3 , n − 1 , the extremal graphs with the maximum distance spectral radius are completely characterized.

Minimizing the least eigenvalue of unicyclic graphs with fixed diameter

Let U(n,d) be the set of unicyclic graphs on n vertices with diameter d. In this article, we determine the unique graph with minimal least eigenvalue among all graphs in U(n,d). It is found that the extremal graph is different from that for the corresponding problem on maximal eigenvalue as done by Liu et al. [H.Q. Liu, M. Lu, F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007) 449-457].

Spectral radius of strongly connected digraphs

Let D be a digraph with vertex set V ( D ) and A be the adjacency matrix of D . The largest eigenvalue of A , denoted by ? ( D ) , is called the spectral radius of the digraph D . In this paper, we establish some sharp upper or lower bounds for digraphs with some given graph parameters, such as clique number, girth, and vertex connectivity, and characterize the corresponding extremal graphs. In addition, we give the exact value of the spectral radii of those digraphs.

On the Laplacian spread of graphs

Abstract The Laplacian spread s ( G ) of a graph G is defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of G . Several upper bounds of Laplacian spread and corresponding extremal graphs are obtained in this paper. Particularly, if G is a connected graph with n ( ≥ 5 ) vertices and m ( n − 1 ≤ m ≤ n + 1 ) edges, then s ( G ) ≤ n − 1 with equality if and only if G is obtained from K 1 , n − 1 by adding m − n + 1 edges.

Sharp bounds on distance spectral radius of graphs

Let D(G) denote the distance matrix of a connected graph G. The largest eigenvalue of D(G) is called the distance spectral radius of a graph G, denoted by ϱ(G). In this article, we give sharp upper and lower bounds for the distance spectral radius and characterize those graphs for which these bounds are best possible.

The base of a primitive, nonpowerful sign pattern with exactly d nonzero diagonal entries

In [J.Y. Shao, L.H. You, H.Y. Shan, Bound on the base of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 2-3, 285-300], Shao, You and Shan extended the concept of the base from powerful sign pattern matrices to nonpowerful (generalized) sign pattern matrices. It is well known that the properties of the power sequences of different classes of sign pattern matrices may be very different. In this paper, we consider the base set of the primitive nonpowerful square sign pattern matrices of order n with exactly d (with d>=1) nonzero diagonal entries. The base set is shown to be {2,3,...,3n-d-1}. The extremal sign pattern matrices with both the least number n+d nonzero entries and the maximum base 3n-d-1 are characterized.