15 research outputs found
Signless Laplacian spectral radii of graphs with given chromatic number
AbstractLet G be a simple graph with vertices v1,v2,…,vn, of degrees Δ=d1⩾d2⩾⋯⩾dn=δ, respectively. Let A be the (0,1)-adjacency matrix of G and D be the diagonal matrix diag(d1,d2,…,dn). Q(G)=D+A is called the signless Laplacian of G. The largest eigenvalue of Q(G) is called the signless Laplacian spectral radius or Q-spectral radius of G. Denote by χ(G) the chromatic number for a graph G. In this paper, for graphs with order n, the extremal graphs with both the given chromatic number and the maximal Q-spectral radius are characterized, the extremal graphs with both the given chromatic number χ≠4,5,6,7 and the minimal Q-spectral radius are characterized as well
Bounds on graph eigenvalues II
Some recent results on graph eigenvalues are improved. In particular, among
all graphs of given order with no cliques of order the -partite
Turan graph has maximal spectral radius
Sharp Bounds for the Signless Laplacian Spectral Radius in Terms of Clique Number
In this paper, we present a sharp upper and lower bounds for the signless
Laplacian spectral radius of graphs in terms of clique number. Moreover, the
extremal graphs which attain the upper and lower bounds are characterized. In
addition, these results disprove the two conjectures on the signless Laplacian
spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the
signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].Comment: 15 pages 1 figure; linear algebra and its applications 201
Linear spectral Turan problems for expansions of graphs with given chromatic number
An -uniform hypergraph is linear if every two edges intersect in at most
one vertex. The -expansion of a graph is the -uniform
hypergraph obtained from by enlarging each edge of with a vertex subset
of size disjoint from the vertex set of such that distinct edges are
enlarged by disjoint subsets. Let and
be the maximum number of edges and the maximum
spectral radius of all -free linear -uniform hypergraphs with
vertices, respectively. In this paper, we present the sharp (or asymptotic)
bounds of and by
establishing the connection between the spectral radii of linear hypergraphs
and those of their shadow graphs, where is a -color critical graph
or a graph with chromatic number
The Laplacian-energy like of graphs
AbstractAssume that μ1,μ2,…,μn are eigenvalues of the Laplacian matrix of a graph G. The Laplacian-energy like of G, is defined as follows: LEL(G)=∑i=1nμi. In this note, we give upper bounds for LEL(G) in terms of connectivity or chromatic number and characterize the corresponding extremal graphs
Disproof of a conjecture on the minimum spectral radius and the domination number
Let be the set of all connected graphs on vertices with
domination number . A graph is called a minimizer graph if it attains
the minimum spectral radius among . Very recently, Liu, Li and
Xie [Linear Algebra and its Applications 673 (2023) 233--258] proved that the
minimizer graph over all graphs in must be a tree.
Moreover, they determined the minimizer graph among
for even , and posed the conjecture on the
minimizer graph among for odd . In this
paper, we disprove the conjecture and completely determine the unique minimizer
graph among for odd
On the distance spectrum and distance energy of complement of subgroup graphs of dihedral group
Let G is a connected simple graph and V(G) = {v1, v2, ..., vp} is vertex set of G. The distance matrix of G is a matrix D(G) = [d ij ] of order p where [d ij ] = d(v i , v j ) is distance between v i and v j in G. The set of all eigenvalues of matrix D(G) together with their corresponding multiplicities is named the distance spectrum of G and denoted by spec D (G). The distance energy of G is , where λi are eigenvalues of D(G). In the recent paper, the distance spectrum and distance energy of complement of subgroup graphs of dihedral group are determined
Q-spectral and L-spectral radius of subgroup graphs of dihedral group
Research on Q-spectral and L-spectral radius of graph has been attracted many attentions. In other hand, several graphs associated with group have been introduced. Based on the absence of research on Q-spectral and L-spectral radius of subgroup graph of dihedral group, we do this research. We compute Q-spectral and L-spectral radius of subgroup graph of dihedral group and their complement, for several normal subgroups. Q-spectrum and Lspectrum of these graphs are also observed and we conclude that all graphs we discussed in this paper are Q-integral dan L-integral