Some properties of Laplacian eigenvalues for generalized star graphs
Sharp bounds for the sum of the squares of the degrees of a graph
Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs
In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with vertices and clique number are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.
Nordhaus-Gaddum-Type Results for Resistance Distance-Based Graph Invariants
Two decades ago, resistance distance was introduced to characterize “chemical distance” in (molecular) graphs. In this paper, we consider three resistance distance-based graph invariants, namely, the Kirchhoff index, the additive degree-Kirchhoff index, and the multiplicative degree-Kirchhoff index. Some Nordhaus-Gaddum-type results for these three molecular structure descriptors are obtained. In addition, a relation between these Kirchhoffian indices is established.
On the energy and spectral properties of the he matrix of hexagonal systems
The He matrix, put forward by He and He in 1989, is designed as a means for uniquely representing the structure of a hexagonal system (= benzenoid graph). Observing that the He matrix is just the adjacency matrix of a pertinently weighted inner dual of the respective hexagonal system, we establish a number of its spectral properties. Afterwards, we discuss the number of eigenvalues equal to zero of the He matrix of a hexagonal system. Moreover, we obtain a relation between the number of triangles...
Chromatic number and some multiplicative vertex-degree-based indices of graphs
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