Spectral radius of graphs forbidden C7C_7 or C6△C_6^{\triangle}

Let $C_k^{\triangle}$ be the graph obtained from a cycle $C_{k}$ by adding a new vertex connecting two adjacent vertices in $C_{k}$. In this note, we obtain the graph maximizing the spectral radius among all graphs with size $m$ and containing no subgraph isomorphic to $C_6^{\triangle}$. As a byproduct, we will show that if the spectral radius $\lambda(G)\ge1+\sqrt{m-2}$, then $G$ must contains all the cycles $C_i$ for $3\le i\le 7$ unless $G\cong K_3\nabla \left(\frac{m-3}{3}K_1\right)$.Comment: 11 pages, 1 figur

Spectral extremal graphs for the bowtie

Let $F_k$ be the (friendship) graph obtained from $k$ triangles by sharing a common vertex. The $F_k$-free graphs of order $n$ which attain the maximal spectral radius was firstly characterized by Cioab\u{a}, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that $n$ is sufficiently large. In this paper, we get rid of the condition on $n$ being sufficiently large if $k=2$. The graph $F_2$ is also known as the bowtie. We show that the unique $n$-vertex $F_2$-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if $n\ge 7$, and the condition $n\ge 7$ is tight. Our result is a spectral generalization of a theorem of Erd\H{o}s, F\"{u}redi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that $\mathrm{ex}(n,F_2)=\left\lfloor {n^2}/{4} \right\rfloor +1$. Moreover, we study the spectral extremal problem for $F_k$-free graphs with given number of edges. In particular, we show that the unique $m$-edge $F_2$-free spectral extremal graph is the join of $K_2$ with an independent set of $\frac{m-1}{2}$ vertices if $m\ge 8$, and the condition $m\ge 8$ is tight.Comment: 22 pages, 6 figures. This is the published versio

Soft tissue modelling and facial movement simulation using the finite element method

This thesis presents a framework for soft tissue modelling, facial surgery simulation, and facial movement synthesis based on the volumetric finite element method. Assessment of facial appearance pre- and post-surgery is of major concern for both patients and clinicians. Pre-surgical planning is a prerequisite for successful surgical procedures and outcomes. Early computer-assisted facial models have been geometrically based. They are computationally efficient, but cannot give an accurate prediction for facial surgery simulation. Therefore, in this thesis, the emphasis is placed on physically-based methods, especially the finite element technique. To achieve realistic surgery simulation, soft tissue modelling is of crucial importance. Thus, in this thesis, considerable effort has been directed to develop constitutive equations for facial skeletal muscles. The skeletal muscle model subsequently developed is able to capture the complex mechanical properties of skeletal muscle, which are active, quasi-incompressible, fibre-reinforced and hyperelastic. In addition, to improve the characterisation of in-vivo muscle behaviour, a technique has been developed to visualise the internal fibre arrangement of skeletal muscle using the FEM-NURBS method, which is the combination of the finite element method and the non-uniform rational B-spline solid mathematical representation. Another principal contribution made in this thesis is the three-dimensional finite element facial model, which can be used for the simulations of facial surgery and facial movement. The procedure of one cranio-facial surgery is simulated by using this facial model and the numerical predictions show a good agreement with the patient post-surgical data. In addition, it would be very helpful to also simulate the facial movement and facial expressions. In this thesis, two facial expressions (smile and disgust) and the mouth opening are simulated to assess the post-surgical appearance and test the muscle-driven facial movement simulation method

Three-dimensional multifractal analysis of trabecular bone under clinical computed tomography

Purpose: An adequate understanding of bone structural properties is critical for predicting fragility conditions caused by diseases such as osteoporosis, and in gauging the success of fracture prevention treatments. In this work we aim to develop multiresolution image analysis techniques to extrapolate high-resolution images predictive power to images taken in clinical conditions. Methods: We performed multifractal analysis (MFA) on a set of 17 ex vivo human vertebrae clinical CT scans. The vertebræ failure loads (FFailure) were experimentally measured. We combined bone mineral density (BMD) with different multifractal dimensions, and BMD with multiresolution statistics (e.g., skewness, kurtosis) of MFA curves, to obtain linear models to predict FFailure. Furthermore we obtained short- and long-term precisions from simulated in vivo scans, using a clinical CT scanner. Ground-truth data - high-resolution images - were obtained with a High-Resolution Peripheral Quantitative Computed Tomography (HRpQCT) scanner. Results: At the same level of detail, BMD combined with traditional multifractal descriptors (Lipschitz-Hölder exponents), and BMD with monofractal features showed similar prediction powers in predicting FFailure (87%, adj. R2). However, at different levels of details, the prediction power of BMD with multifractal features raises to 92% (adj. R2) of FFailure. Our main finding is that a simpler but slightly less accurate model, combining BMD and the skewness of the resulting multifractal curves, predicts 90% (adj. R2) of FFailure. Conclusions: Compared to monofractal and standard bone measures, multifractal analysis captured key insights in the conditions leading to FFailure. Instead of raw multifractal descriptors, the statistics of multifractal curves can be used in several other contexts, facilitating further research.Fil: Baravalle, Rodrigo Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; ArgentinaFil: Thomsen, Felix Sebastian Leo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Sur; ArgentinaFil: Delrieux, Claudio Augusto. Universidad Nacional del Sur; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Lu, Yongtao. Dalian University of Technology; ChinaFil: Gómez, Juan Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; ArgentinaFil: Stošić, Borko. Universidade Federal Rural Pernambuco; BrasilFil: Stošić, Tatijana. Universidade Federal Rural Pernambuco; Brasi

Relationship between the morphological, mechanical and permeability properties of porous bone scaffolds and the underlying microstructure

Bone scaffolds are widely used as one of the main bone substitute materials. However, many bone scaffold microstructure topologies exist and it is still unclear which topology to use when designing scaffold for a specific application. The aim of the present study was to reveal the mechanism of the microstructure-driven performance of bone scaffold and thus to provide guideline on scaffold design. Finite element (FE) models of five TPMS (Diamond, Gyroid, Schwarz P, Fischer-Koch S and F-RD) and three traditional (Cube, FD-Cube and Octa) scaffolds were generated. The effective compressive and shear moduli of scaffolds were calculated from the mechanical analysis using the FE unit cell models with the periodic boundary condition. The scaffold permeability was calculated from the computational fluid dynamics (CFD) analysis using the 4×4×4 FE models. It is revealed that the surface-to-volume ratio of the Fischer-Koch S-based scaffold is the highest among the scaffolds investigated. The mechanical analysis revealed that the bending deformation dominated structures (e.g., the Diamond, the Gyroid, the Schwarz P) have higher effective shear moduli. The stretching deformation dominated structures (e.g., the Schwarz P, the Cube) have higher effective compressive moduli. For all the scaffolds, when the same amount of change in scaffold porosity is made, the corresponding change in the scaffold relative shear modulus is larger than that in the relative compressive modulus. The CFD analysis revealed that the structures with the simple and straight pores (e.g., Cube) have higher permeability than the structures with the complex pores (e.g., Fischer-Koch S). The main contribution of the present study is that the relationship between scaffold properties and the underlying microstructure is systematically investigated and thus some guidelines on the design of bone scaffolds are provided, for example, in the scenario where a high surface-to-volume ratio is required, it is suggested to use the Fischer-Koch S based scaffold

Predictive assembling model reveals the self-adaptive elastic properties of lamellipodial actin networks for cell migration

Branched actin network supports cell migration through extracellular microenvironments. However, it is unknown how intracellular proteins adapt the elastic properties of the network to the highly varying extracellular resistance. Here we develop a three-dimensional assembling model to simulate the realistic self-assembling process of the network by encompassing intracellular proteins and their dynamic interactions. Combining this multiscale model with finite element method, we reveal that the network can not only sense the variation of extracellular resistance but also self-adapt its elastic properties through remodeling with intracellular proteins. Such resistance-adaptive elastic behaviours are versatile and essential in supporting cell migration through varying extracellular microenvironments. The bending deformation mechanism and anisotropic Poisson’s ratios determine why lamellipodia persistently evolve into sheet-like structures. Our predictions are confirmed by published experiments. The revealed self-adaptive elastic properties of the networks are also applicable to the endocytosis, phagocytosis, vesicle trafficking, intracellular pathogen transport and dendritic spine formation

Observation of Temperature-Induced Crossover to an Orbital-Selective Mott Phase in Ax_{x}Fe2−y_{2-y}Se2_2 (A=K, Rb) Superconductors

In this work, we study the A$_{x}$Fe$_{2-y}$Se$_2$ (A=K, Rb) superconductors using angle-resolved photoemission spectroscopy. In the low temperature state, we observe an orbital-dependent renormalization for the bands near the Fermi level in which the dxy bands are heavily renormliazed compared to the dxz/dyz bands. Upon increasing temperature to above 150K, the system evolves into a state in which the dxy bands have diminished spectral weight while the dxz/dyz bands remain metallic. Combined with theoretical calculations, our observations can be consistently understood as a temperature induced crossover from a metallic state at low temperature to an orbital-selective Mott phase (OSMP) at high temperatures. Furthermore, the fact that the superconducting state of A$_{x}$Fe$_{2-y}$Se$_2$ is near the boundary of such an OSMP constraints the system to have sufficiently strong on-site Coulomb interactions and Hund's coupling, and hence highlight the non-trivial role of electron correlation in this family of iron superconductors