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The girth of a graph is defined as the length of a shortest cycle in the graph. A $(k; g)$-cage is a graph of minimum order among all $k$-regular graphs with girth $g$. A cycle $C$ in a graph $G$ is termed nonseparating if the graph $G-V(C)$ remains connected. A conjecture, proposed in [T. Jiang, D. Mubayi. Connectivity and Separating Sets of Cages. J. Graph Theory 29(1)(1998) 35--44], posits that every cycle of length $g$ within a $(k; g)$-cage is nonseparating. While the conjecture has been proven for even $g$ in the aforementioned work, this paper presents a proof demonstrating that the conjecture holds true for odd $g$ as well. Thus, the previously mentioned conjecture was proven to be true., Comment: This paper was published in Chinese on April 25, 2001, in Volume 14, Issue 2 of the "MATHEMATICA APPLICATA" on pages 99 to 102. This is the English version of the paper, incorporating additional details and rectifying certain typographical errors

For a given smooth convex cone in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$ which is centered at the origin, we investigate the evolution of strictly mean convex hypersurfaces, which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly, along an inverse curvature flow with the speed equal to $\left(f(r)H\right)^{-1}$, where $f$ is a positive function of the radial distance parameter $r$ and $H$ is the mean curvature of the evolving hypersurfaces. The evolution of those hypersurfaces inside the cone yields a fully nonlinear parabolic Neumann problem. Under suitable constraints on the first and the second derivatives of the radial function $f$, we can prove the long-time existence of this flow, and moreover the evolving hypersurfaces converge smoothly to a piece of the round sphere., Comment: 17 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2104.08884, arXiv:2104.10600, arXiv:2106.05973

In this paper, by imposing suitable assumptions on the weighted function, (under the constraint of fixed weighted volume) a Brock-type isoperimetric inequality for Steklov-type eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven. This conclusion is actually an interesting extension of F. Brock's classical result about the isoperimetric inequality for Steklov eigenvalues of the Laplacian given in the influential paper [Z. Angew. Math. Mech. 81 (2001) 69-71]. Besides, a related open problem has also been proposed in this paper., Comment: 18 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2403.08070

In this paper, by mainly using the rearrangement technique and suitably constructing trial functions, under the constraint of fixed weighted volume, we can successfully obtain several isoperimetric inequalities for the first and the second Dirichlet eigenvalues, the first nonzero Neumann eigenvalue of the Witten-Laplacian on bounded domains in space forms. These spectral isoperimetric inequalities extend those classical ones (i.e. the Faber-Krahn inequality, the Hong-Krahn-Szeg\H{o} inequality and the Szeg\H{o}-Weinberger inequality) of the Laplacian., Comment: 31 pages. Comments are welcome. Several typos have been corrected to v1

An isoperimetric inequality for lower order nonzero Neumann eigenvalues of the Witten-Laplacian on bounded domains in a Euclidean space or a hyperbolic space has been proven in this paper. About this conclusion, we would like to point out two things: It strengthens the well-known Szeg\H{o}-Weinberger inequality for nonzero Neumann eigenvalues of the classical free membrane problem given in [J. Rational Mech. Anal. 3 (1954) 343-356] and [J. Rational Mech. Anal. 5 (1956) 633-636]; Recently, Xia-Wang [Math. Ann. 385 (2023) 863-879] gave a very important progress to the celebrated conjecture of M. S. Ashbaugh and R. D. Benguria proposed in [SIAM J. Math. Anal. 24 (1993) 557-570]. It is easy to see that our conclusion here covers Xia-Wang's this progress as a special case. In this paper, we have also proposed two open problems which can be seen as a generalization of Ashbaugh-Benguria's conjecture mentioned above., Comment: 19 pages. Comments are welcome. Several typos have been corrected to v1. A new reference has been added

Given an eigenvalue $\lambda$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $\lambda=\lambda_{k}=\dots = \lambda_{k+m-1}$, we characterise the lowest and highest orders in the set $\left\{k,\dots,k+m-1\right\}$ for which P\'{o}lya's conjecture holds and fails. In particular, we show that P\'{o}lya's conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when $n$ is greater than two. We further derive P\'{o}lya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li-Yau inequalities for $\mathbb{S}^2$ and $\mathbb{S}^4$, respectively., Comment: 54 pages. Some misprints corrected. To appear in the Annales de l'Institut Fourier

Let $\Omega$ be a bounded domain (with smooth boundary) on the hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of $k$-convex solutions to a class of Hessian quotient equations defined over $\Omega\subset\mathscr{H}^{n}(1)$ and with the vanishing Dirichlet boundary condition., Comment: 13 pages. We have corrected a mistake in the proof of main theorem of the previous version v1 and we have also simplified the proof a lot. Besdies, several new references have been added. Comments are welcome

In this paper, under suitable settings, we can obtain the existence and uniqueness of solutions to a class of Hessian quotient equations with Dirichlet boundary condition in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, which can be seen as a prescribed curvature problem and a continuous work of [12]., Comment: 27 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2111.01345

In this paper, under suitable settings, we can obtain the existence of solutions to a class of prescribed Weingarten curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of closed hypersurface (which is graphic with respect to the base manifold and whose $k$-th Weingarten curvature satisfies some constraint) in a given warped product manifold of special type can be assured., Comment: 19 pages. Two minor typos in Introduction (of v1) have been corrected, and the latest status of reference [2] has been updated. Comments are welcome

In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed $2$-th Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane $\mathscr{H}^{n}(1)\subset\mathbb{R}^{n+1}_{1}$ of center at origin and radius $1$, can be proven., Comment: 19 pages. Comments are welcome

In this paper, we consider the evolution of spacelike graphic curves defined over a piece of hyperbola $\mathscr{H}^{1}(1)$, of center at origin and radius $1$, in the $2$ dimensional Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic curves converge smoothly to a piece of hyperbola of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of $\mathscr{H}^{1}(1)$, as time tends to infinity., Comment: This work has been announced in our previous work [arXiv:2106.05973]. 18 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2104.10600, arXiv:2104.08884

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along the inverse Gauss curvature flow (i.e., the evolving speed equals the $(-1/n)$-th power of the Gaussian curvature) with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of the spacelike graph of a positive constant function defined over the piece of $\mathscr{H}^{n}(1)$ as time tends to infinity., Comment: This work has been announced in our previous work [arXiv:2104.10600v4]. 31 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:2106.05973

In this paper, we consider the evolution of spacelike graphic hypersurfaces defined over a convex piece of hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic hypersurfaces converge smoothly to a piece of hyperbolic plane of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of $\mathscr{H}^{n} (1)$, as time tends to infinity. Clearly, this conclusion is an extension of our previous work [2]., Comment: 16 pages. Our related works (i.e., references [3,4,5] here) will also be put on arXiv soon. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2104.10600, arXiv:2104.08884