Muhuo Liu

A note on sum of powers of the Laplacian eigenvalues of graphs

Abstract For a graph G and a real number α ≠ 0 , the graph invariant s α ( G ) is the sum of the α th power of the non-zero Laplacian eigenvalues of G . This note presents some bounds for s α ( G ) in terms of the vertex degrees of G , and a relation between s α ( G ) and the first general Zagreb index, which is a useful topological index and has important applications in chemistry.

Complete split graph determined by its (signless) Laplacian spectrum

A complete split graph C S ( n , α ) , is a graph on n vertices consisting of a clique on n - α vertices and an independent set on the remaining α ( 1 ? α ? n - 1 ) vertices in which each vertex of the clique is adjacent to each vertex of the independent set. In this paper, we prove that C S ( n , α ) is determined by its Laplacian spectrum when 1 ? α ? n - 1 , and C S ( n , α ) is also determined by its signless Laplacian spectrum when 1 ? α ? n - 1 and α ? 3 .

Kite graphs determined by their spectra

Abstract A kite graph Ki n ,  ω is a graph obtained from a clique K ω and a path P n − ω by adding an edge between a vertex from the clique and an endpoint from the path. In this note, we prove that K i n , n − 1 is determined by its signless Laplacian spectrum when n ≠ 5 and n ≥ 4, and K i n , n − 1 is also determined by its distance spectrum when n ≥ 4.

On partitions of graphs under degree constraints

Abstract Let s , t be two integers, and let g ( s , t ) denote the minimum integer such that the vertex set of a graph of minimum degree at least g ( s , t ) can be partitioned into two nonempty sets which induce subgraphs of minimum degree at least s and t , respectively. In this paper, it is shown that, (1) for positive integers s and t , g ( s , t ) ≤ s + t on ( K 4 − e ) -free graphs except K 3 , and (2) for integers s ≥ 2 and t ≥ 2 , g ( s , t ) ≤ s + t − 1 on triangle-free graphs in which no two quadrilaterals share edges. Our first conclusion generalizes a result of Kaneko (1998), and the second generalizes a result of Diwan (2000).

On the ordering of distance-based invariants of graphs

Let d(u, v) be the distance between u and v of graph G, and let Wf(G) be the sum of f(d(u, v)) over all unordered pairs {u, v} of vertices of G, where f(x) is a function of x. In some literatures, Wf(G) is also called the Q-index of G. In this paper, some unified properties to Q-indices are given, and the majorization theorem is illustrated to be a good tool to deal with the ordering problem of Q-index among trees with n vertices. With the application of our new results, we determine the four largest and three smallest (resp. four smallest and three largest) Q-indices of trees with n vertices for strictly decreasing (resp. increasing) nonnegative function f(x), and we also identify the twelve largest (resp. eighteen smallest) Harary indices of trees of order n ≥ 22 (resp. n ≥ 38) and the ten smallest hyper-Wiener indices of trees of order n ≥ 18, which improve the corresponding main results of Xu (2012) and Liu and Liu (2010), respectively. Furthermore, we obtain some new relations involving Wiener index, hyper-Wiener index and Harary index, which gives partial answers to some problems raised in Xu (2012).