It was recently noticed that lemniscates do not survive Laplacian growth [12] (2010). This raises the question: “Is there a growth process for which polynomial lemniscates are solutions?” The answer is “yes”, and the law governing the boundary velocity is reciprocal to that of Laplacian growth. Viewing lemniscates as solutions to a moving-boundary problem gives a new perspective on results from...

A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main. Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In the process, infinitely many non-is...

We discuss the relationships between the notion of intrinsic ultracontractivity, parabolic Harnack principle, compactness of the 1-resolvent of the Neumann Laplacian, and non-trap property for Euclidean domains with finite Lebesgue measure. In particular, we give an answer to an open problem raised by Davies and Simon in 1984 about the possible relationship between intrinsic ultracontractivity ...

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 Lapla...

Let G be a simple connected graph. The transmission of any vertex v of a graph G is defined as the sum of distances of a vertex v from all other vertices in a graph G. Then the distance signless Laplacian matrix of G is defined as D^{Q}(G)=D(G)+Tr(G), where D(G) denotes the distance matrix of graphs and Tr(G) is the diagonal matrix of vertex transmissions of G. For a given minimum dominating se...

In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, where k ≥ 3, reaches its upper bound 2∆(G), where ∆(G) is the largest degree of G, if and only if G is regular. Thus the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and oddbipartite. We show that an s-cycle G, as a k-uniform hypergraph...

This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris’ conjecture [The Laplacian spectrum of graph...

For a simple connected graph G with n-vertices having Laplacian eigenvalues μ1, μ2, . . . , μn−1, μn = 0, and signless Laplacian eigenvalues q1, q2, . . . , qn, the Laplacian-energy-like invariant(LEL) and the incidence energy (IE) of a graph G are respectively defined as LEL(G) = ∑n−1 i=1 √ μi and IE(G) = ∑n i=1 √ qi. In this paper, we obtain some sharp lower and upper bounds for the Laplacian...

We consider the problem of determining the Q–integral graphs, i.e. the graphs with integral signless Laplacian spectrum. First, we determine some infinite series of such graphs having the other two spectra (the usual one and the Laplacian) integral. We also completely determine all (2, s)–semiregular bipartite graphs with integral signless Laplacian spectrum. Finally, we give some results conce...