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A 2-edge-connected graph G is minimally 2-edge-connected if deleting arbitrary edge of G always leaves a graph which is not 2-edge-connected. In this paper, we completely characterize the (minimally) 2-edge-connected graphs with given size having the maximal signless Laplacian (Laplacian) index. [ABSTRACT FROM AUTHOR]

In this paper, we study the signless Laplacian spectral Turán type problem in terms of the size, obtain sharp upper bounds on the signless spectral radius of C 3 ( C 4 )-free graphs with given size and minimum degree δ ≥ 2 , and characterize the corresponding extremal graphs completely. [ABSTRACT FROM AUTHOR]

A c-cyclic graph is a connected graph with n vertices and n + c−1 edges. In this paper, we consider the problem that: in the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices, which graph has the largest A α -spectral radius. We shall show that, for 0 ≤ c ≤ n − 4 , the extremal graph must be a graph with an isolated vertex. This implies that the maximum degree of G is larger, then the A α -spectral radius of its complement graph is also larger. We also determine the structure of the extremal graph with largest A α -spectral radius among the class of graphs, each of which is the complement graph of a c-cyclic graph with n vertices and maximum degree n−1 for 0 ≤ c ≤ n − 3 and n−2 for 0 ≤ c ≤ n − 4 , respectively. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. For real number α ∈ [0, 1], Nikiforov defined the Aα-matrix of G as Aα(G) = αD(G) + (1 − α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. In this paper, we obtain a sharp upper bound on the largest eigenvalue ρα(G) of Aα(G) for α ∈ [1/2, 1). Employing this upper bound, we prove that 'For connected G1 and G2 with n vertices and m edges, if the maximum degree Δ(G1) ≥ 2α(1 − α)(2m − n + 1) + 2α and Δ(G1) > Δ(G2), then ρ α (G 1) > ρ α (G 2) '. Let λα(G) denote the least eigenvalue of Aα(G). For α ∈ (1/2, 1), we prove that 'For two connected G1 and G2, if the minimum degree δ (G 1) ≤ 1 1 − α − 2 and δ(G1) < δ(G2), then λα(G1) < λα(G2)'. [ABSTRACT FROM AUTHOR]

Let g (G) denote the girth of a graph G. In this paper, we determine the unique graph with the maximum least Q-eigenvalue among all unicyclic graphs of order n = 6k (k ≥ 8) with perfect matchings. For the cases when n = 6k + 2 and n = 6k + 4, we prove that g (G) = 3 if G is a graph with the maximum least Q-eigenvalue, and provide a conjecture and a problem on the sharp upper bound of the least Q-eigenvalue. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph. For any real number α ∈ [ 0 , 1 ] , Nikiforov defined the A α -matrix of G as A α (G) = α D (G) + (1 - α) A (G) , where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. The largest eigenvalue of A α (G) is called the A α -spectral radius of G. In this paper, we give sharp upper bounds on the A α -spectral radius of C 4 -free graphs and Halin graphs for α ∈ [ 1 / 2 , 1) respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the algebraic connectivity of a Hamiltonian graph, and determine all Hamiltonian graphs whose algebraic connectivity attain the minimum among all Hamiltonian graphs onnvertices. [ABSTRACT FROM AUTHOR]

TheQ-indexof graphGis the largest eigenvalueq(G)of its signless Laplacian. In this paper, we prove that the graphhas the maximalQ-index among all outer-planar graphs of ordern. [ABSTRACT FROM PUBLISHER]

Objective: The aim of this study was to assess the therapeutic effects and complications of tension band wiring (TBW) through the use of double-cannulated screws versus conventional TBW in the treatment of olecranon fractures., Methods: Eligible participants were randomly assigned to 2 groups to undergo different methods of fixation. The related indices and data of the 2 groups were collected for comparative analysis after an average follow-up of 32.7±6.6 months., Results: Average fracture healing time was 11.4 weeks in the double-screw TBW group and 12.6±1.8 weeks in the conventional TBW group (p=0.000). There was significant difference in complications related to fixation between the 2 groups. In the double-screw TBW group (42 patients), 2 patients felt screw head prominence with no pain and requested no further intervention; in contrast, 21 patients experienced complications associated with internal fixation in the conventional TBW group. Mean Mayo Elbow Performance Score (MEPS) score was 87.90±6.0 in the double-screw TBW group, compared to 83.67±6.6 in the conventional TBW group at 24-month follow-up (p=0.002). The rate of elbow function in the double-screw TBW group (29/42, 69.05%) was higher than that of the conventional TBW group (16/46, 34.78%) (p=0.000)., Conclusion: In comparison with conventional TBW, TBW with double-cannulated screws can significantly reduce complications, lower reoperation rate, improve elbow function, shorten healing time, as well as diminish surgical trauma.