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Let q(G) denote the signless Laplacian spectral radius of a graph G. In this paper, we first give an upper bound on q(G) of a connected graph Gwith fixed size m≥3k(k∈Z+)and maximum degree Δ≤m-k. For two connected graphs G1and G2with size m≥4, employing this upper bound, we prove that q(G1)>q(G2)if Δ(G1)>Δ(G2)and Δ(G1)≥2m3+1. As an application, we determine the first ⌊d/2⌋graphs with the largest signless Laplacian spectral radius among all graphs with fixed size and diameter.

A single-cone tree (unicyclic graph) is the join of a complete graph K1and a tree (unicyclic graph). Suppose π=(d1,d2,…,dn)and π′=(d1′,d2′,…,dn′)are two non-increasing degree sequences. We say πis majorizated by π′, denoted by π⊲π′, if and only if π≠π′, ∑i=1ndi=∑i=1ndi′, and ∑i=1jdi≤∑i=1jdi′for all j=1,2,…,n-1. We use Jπto denote the class of single-cone trees (unicyclic graphs) with degree sequence π. Suppose that πand π′are two different non-increasing degree sequences of single-cone trees (unicyclic graphs). Let ρand ρ′be the largest spectral radius of the graphs in Jπand Jπ′, respectively, μand μ′be the largest signless Laplacian spectral radius of the graphs in Jπand Jπ′, respectively. In this paper, we prove that if π⊲π′, then ρ<ρ′and μ<μ′.

The Laplacian spread of a graph Gis defined to be the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of G. Let $${\mathcal {Q}}_t(n, d)=\{G\, |\, G-v_0\ \hbox {is a tree on }n-1\hbox { vertices and } d_G(v_0)=d\}.$$Qt(n,d)={G|G-v0is a tree onn-1vertices anddG(v0)=d}.Recently, Y. Xu and J. Meng characterized the unique graph with maximum Laplacian spread among all graphs in the set $${\mathcal {Q}}_t(n, d)$$Qt(n,d)with $$1\le d\le (n-4)/2$$1≤d≤(n-4)/2. In this paper, we extend their result by determining the unique graph with maximum Laplacian spread among all graphs in the set $${\mathcal {Q}}_t(n, d)$$Qt(n,d)with $$1\le d\le n-5$$1≤d≤n-5.

The under surface depression of foam glass blanks as one of the typical defects, influences and impedes the quality of foam glass and material cutting rate. This paper studies the water content of release agent and open flame burning temperature effects on under depression of foam glass, to reveal its generation mechanism, using TG-DSC, XRD, FTIR, and other analytical methods. The results show that, with the increase of drying temperature, the under surface depression phenomenon reduces, indicating the air cavity caused by heating vaporization of water molecules. The surface is white. In the early flame heating, firing temperature is more than 700°C, by the mold bottom heating carbon, resulting in carbon black early reaction gas, not be wrapped by glass, eventually gathering to form an air chamber. The surface is grey.

This study selected high alkali aluminosilicate glass as the research object. It’s belongs to the SiO2-Al2O3-RO-R2O glass system. By changing the content of Al2O3 in the SiO2-Al2O3-RO-R2O glass system, designed five basic glass formula. Using conventional melting method prepared the high alkali aluminosilicate glass. In the case of the same heat treatment, the glass formulations are the most important factor in the glass surface tension. Using the glass surface tension measuring instrument, focus on the content of Al2O3 on the glass surface tension’s effects law. The results show that the high alkali aluminosilicate glass’s surface tension are increase with the increase of Al2O3 content. The relation between glass surface tension and the content of Al2O3 conforms to the formula Y=290+6.5WAl2O3, Al2O3 factors influence the surface tension of the glass is 6.5.

At high temperature conditions, the glass molten by the combined action of gravity and surface tension form an oval. Through digital image measurement system we can obtain glass molten oval contours and the contact angle. By the Bashforth-Adams equation which deduced from Young- Laplace equation to obtain the glass molten' surface tension at the corresponding temperature conditions. This method has good reproducibility and accuracy.