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Given a simple graph G, its Laplacian-energy-like invariant LEL(G) and incidence energy IE(G) are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. This paper obtains some improved bounds on LEL and IE of the 𝓡-graph and 𝓠-graph for a regular graph. Theoretical analysis indicates that these results improve some known results. In addition, some new lower bounds on LEL and IE of the line graph of a semiregular graph are also given.

Precious metal Pt‐based electrocatalysts have been widely used in the methanol catalytic oxidation anodes in direct methanol fuel cells. However, decreasing their cost and improving their efficiency and durability have still been challenging. Herein, ternary PtPdCu nanocatalysts are synthesized through a facile one‐step hydrothermal synthesis method. When KI is present with a suitable amount in the synthesis, PtPdCu nanospheres with surface‐embedded CuI clusters (CuI/PtPdCu) are fabricated. However, without KI, the prepared PtPdCu catalysts show a distinct hollow structure (h‐PtPtCu). CuI/PtPdCu displays the highest specific activity with enhancement 4 times higher than commercial Pt/C for the methanol oxidation reaction in an alkaline medium. The superior activity can be attributed to two aspects: 1) the electronic effect originating from the highly alloyed PtPdCu; 2) the synergetic effect resulting from surface inlaid CuI clusters, which can promote the CO intermediate removal. Furthermore, because of the stable Pt–Pd‐rich surface and its special linked hollow structure, the h‐PtPtCu catalyst exhibits good durability with only a 3.6% decay in the specific activity. [ABSTRACT FROM AUTHOR]

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. Recently, there has been a renewed interest in studying Seidel spectrum of graphs and some achievements have been made in this regard. In this paper, we determine the Seidel spectra of some variants of corona operations, such as edge corona, subdivision-vertex corona, subdivision-vertex neighbourhood corona of two graphs and so on. The corresponding Seidel eigenvectors are also described completely. Applying the obtained results, we give some sufficient and necessary conditions for edge corona, subdivision-vertex corona and subdivision-vertex neighbourhood corona of two graphs to be Seidel integral. [ABSTRACT FROM AUTHOR]

The Seidel spectrum of a graph is defined as the multiset of all eigenvalues of its Seidel matrix. For two simple connected graphs G 1 and G 2 , let us denote the subdivision-vertex join and subdivision-edge join by G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 , respectively. In this paper, we completely determine the Seidel spectrum and corresponding Seidel eigenvectors of G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 . As an application, we give a sufficient and necessary condition for G 1 ∨ ̇ G 2 and G 1 ⊻ G 2 to be Seidel integral. [ABSTRACT FROM AUTHOR]

Let S(G) be the Seidel matrix of a graph G. The Seidel energy of G, denoted by ES(G), is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix S(G) of G. In this paper, we consider the change of Seidel energy of graphs due to edge deletion. It is proved that the Seidel energy of the tripartite Turán graph T(n, 3) of order n ≥ 9 is always increased when an edge is deleted. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with vertex set V (G) = { v 1 , v 2 , ... , v n } and edge set E (G). The arithmetic–geometric matrix A AG (G) = (h i j) of G is a matrix of order n defined by h i , j = 1 2 d i d j + d j d i if v i v j ∈ E (G) and 0 otherwise, where d i stands for the degree of the vertex v i in G. The arithmetic–geometric characteristic polynomial of G is the characteristic polynomial of A AG (G). The arithmetic–geometric energy ℰ AG (G) of G is the sum of absolute values of all eigenvalues of A AG (G). In this paper, we obtain the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy of some specific graphs. In addition, we also consider the arithmetic–geometric characteristic polynomial and arithmetic–geometric energy change of these graphs when one edge is deleted. [ABSTRACT FROM AUTHOR]

For two graphs G and H, the corona product G ∘ H is the graph obtained by taking one copy of G and | V G | copies of H, and joining the ith vertex of G with every vertex of the ith copy of H. In this paper, we study the state transfer of corona relative to the signless Laplacian matrix. We explore some conditions that guarantee the signless Laplacian perfect state transfer in G ∘ H . We prove that G ∘ K m has no signless Laplacian perfect state transfer for some special m. We also show that K 2 ∘ H has pretty good state transfer but no perfect state transfer relative to the signless Laplacian matrix for a regular graph H. Furthermore, we show that n K 2 ¯ ∘ K 1 has signless Laplacian pretty good state transfer, where n K 2 ¯ is the cocktail party graph. [ABSTRACT FROM AUTHOR]

Let G [ F, V k , H v ] be the graph with k pockets, where F is a simple graph of order n  ≥ 1, V k = { v 1 , … , v k } is a subset of the vertex set of F and H v is a simple graph of order m  ≥ 2, v is a specified vertex of H v . Also let G [ F, E k , H uv ] be the graph with k edge-pockets, where F is a simple graph of order n  ≥ 2, E k = { e 1 , … , e k } is a subset of the edge set of F and H uv is a simple graph of order m  ≥ 3, uv is a specified edge of H uv such that H u v − u is isomorphic to H u v − v . In this paper, we obtain some results describing the signless Laplacian spectra of G [ F, V k , H v ] and G [ F, E k , H uv ] in terms of the signless Laplacian spectra of F, H v and F, H uv , respectively. In addition, we also give some results describing the adjacency spectrum of G [ F, V k , H v ] in terms of the adjacency spectra of F, H v . Finally, as many applications of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral 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]

We establish an inclusion relation between two known inclusion intervals of matrix singular values in some special case. In addition, based on the use of positive scale vectors, a known inclusion interval of matrix singular values is also improved. [ABSTRACT FROM AUTHOR]

Abstract: Let D be a digraph of order n and denote all the eigenvalues of the skew-adjacency matrix of D. The skew energy of D is defined as . In this paper, it is proved that for any positive integer , there exists a k-regular graph of order n having an orientation D with . This work positively answers a problem proposed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin So, The skew energy of a digraph, Linear Algebra Appl. 432 (2010) 1825–1835]. In addition, a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of when the sub-direct sum of two strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix is studied. In particular, it is shown that the subdirect sum of overlapping principal submatrices of strictly diagonally dominant P-matrices is a strictly diagonally dominant P-matrix. It is also established that the 2-subdirect sum of two totally nonnegative matrices is a totally nonnegative matrix under some conditions. It is obtained that a partial totally nonnegative matrix, whose graph of the specified entries is a monotonically labeled 2-chordal graph, has a totally nonnegative completion. Finally, a positive answer to the question (IV) in Fallat and Johnson [Shaun M. Fallat, C.R. Johnson, J.R. Torregrosa, A.M. Urbano, P-matrix completions under weak symmetry assumptions, Linear Algebra Appl. 312 (2000) 73–91] is given for P 0-matrices. [Copyright &y& Elsevier]

With the expansion of global energy problems and the deepening of research on oxygen reduction reaction (ORR) in alkaline media, the development of low cost and high electrocatalytic performance catalysts has become a research hotspot. In this study, a hexagonal Pd-C-MnOOH composite catalyst was prepared by using the triblock copolymer P123 as the reducing agent and protective agent, sucrose as the carbon source and self-made MnOOH as the carrier under hydrothermal conditions. When the Pd load is 20% and the C/MnOOH ratio is 1 : 1, the 20% Pd-C-MnOOH-1 : 1 catalyst obtained by the one-step method has the highest ORR activity and stability in the alkaline system. At 1600 rpm, the limiting diffusion current density and half-wave potential of the 20% Pd-C-MnOOH-1 : 1 electrocatalyst are -4.78 mA cm -2 and 0.84 V, respectively, which are better than those of the commercial 20%Pd/C catalyst. According to the Koutecky-Levich (K-L) equation and the linear fitting results, the electron transfer number of the 20%Pd-C-MnOOH-1 : 1 electrocatalyst for the oxygen reduction reaction is 3.8, which is similar to that of a 4-electron process. After 1000 cycles, the limiting diffusion current density of the 20%Pd-C-MnOOH-1 : 1 catalyst is -4.61 mA cm -2 , which only decreases by 3.7%, indicating that the 20%Pd-C-MnOOH-1 : 1 catalyst has good stability. The reason for the improvement of the ORR performance of the Pd-C-MnOOH composite catalyst is the improvement of the conductivity of the carbon layer formed by original carbonization, the regular hexagonal highly active Pd particles and the synergistic catalytic effect between Pd and MnOOH. The method of introducing triblock copolymers in the synthesis of oxides and metal-oxide composite catalysts is expected to be extended to other electrocatalysis fields.