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Reducing the amount of precious metal Pt in proton exchange membrane fuel cells while improving its electrocatalytic activity is one of the core problems in this field. The formation of stable ultrafine Pt nanoparticles is one of the effective methods for improving the utilization rate of Pt and indirectly reducing the amount of Pt. The traditional carbon support has carbon corrosion phenomenon, and the exploration of suitable catalyst support can also improve the utilization rate of the Pt unit and indirectly reduce its consumption. In this paper, ultrafine Pt nanoparticles (2 nm) have been loaded on white carbon black (WCB) in situ by a simple hydrothermal reaction. When the Pt mass load is 40%, the Pt particles are highly dispersed on the WCB, but when the load is increased to 60%, the Pt particles will agglomerate significantly. Due to its high dispersion and ultrafine particle size, 40%Pt/SiO2 catalyst shows enhanced methanol oxidation reaction (MOR) performance in acidic electrolyte. The maximum oxidation peak current density (163.3mA mg−1Pt) of the MOR is superior to that of the commercial carbon-supported Pt catalyst. This study not only describes the design of a highly dispersed, ultrafine particle size of Pt-based catalyst, but also provides a new choice for a new catalyst carrier. [ABSTRACT FROM AUTHOR]

A mixed graph M G is the graph obtained from an unoriented simple graph G by giving directions to some edges of G, where G is often called the underlying graph of M G . In this paper, we discuss the determinants of two classes of incidence matrices of the second kind of rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of M G . Moreover, we give two sufficient conditions that the module of any cofactor of Hermitian (quasi-)Laplacian matrix of the second kind of the mixed graph M G is equal to the number of spanning trees of the underlying graph G. [ABSTRACT FROM AUTHOR]

Given two graphs G 1 of order n 1 and G 2 , the neighborhood corona of G 1 and G 2 , denoted by G 1 ★ G 2 , is the graph obtained by taking one copy of G 1 and taking n 1 copies of G 2 , in the meanwhile, linking all the neighbors of the i-th vertex of G 1 with all vertices of the i-th copy of G 2 . In our work, we give some conditions that G 1 ★ G 2 is not periodic. Furthermore, we demonstrate some sufficient conditions for G 1 ★ G 2 having no perfect state transfer. Some examples are provided to explain our results. In addition, for the reason that the graph admitting perfect state transfer is rare, we also consider pretty good state transfer on neighborhood corona of two graphs. We show some sufficient conditions for G 1 ★ G 2 admitting pretty good state transfer. [ABSTRACT FROM AUTHOR]

For a simple connected graph G, the Q-generating function of the numbers N k of semi-edge walks of length k in G is defined by W Q (t) = ∑ k = 0 ∞ N k t k . This paper reveals that the Q-generating function W Q (t) may be expressed in terms of the Q-polynomials of the graph G and its complement G ¯ . Using this result, we study some Q-spectral properties of graphs and compute the Q-polynomials for some graphs obtained from various graph operations, such as the complement graph of a regular graph, the join of two graphs and the (edge)corona of two graphs. As another application of the Q-generating function W Q (t) , we also give a combinatorial interpretation of the Q-coronal of G, which is defined to be the sum of the entries of the matrix (λ I n - Q (G)) - 1 . This result may be used to obtain the many alternative calculations of the Q-polynomials of the (edge)corona of two graphs. Further, we also compute the Q-generating functions of the join of two graphs and the complete multipartite graphs. [ABSTRACT FROM AUTHOR]

Many variants of join operations of graphs have been introduced, and their spectral properties have been studied extensively by many researchers. This paper mainly focuses on the Laplacian spectra of some double join operations of graphs. We first introduce the conception of double join matrix and provide a complete information about its eigenvalues and the corresponding eigenvectors. Further, we define four variants of double join operations based on subdivision graph, Q-graph, R-graph and total graph. Applying the result obtained about double join matrices, we give an explicit complete characterization of the Laplacian eigenvalues and the corresponding eigenvectors of four variants in terms of the Laplacian eigenvalues and the eigenvectors of factor graphs. These results generalize some well-known results on the Laplacian spectra of some join operations of graphs. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph of order n with degree sequence ( d, d, ..., d). Denote ( t) = Σ d, ( m) = ( t)/ d and ( N) = Σ ( t), where α is a real number. Denote by λ( G) and µ( G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present some upper and lower bounds of λ( G) and µ( G) in terms of ( t), ( m) and ( N). Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results. [ABSTRACT FROM AUTHOR]