Local cohomology modules of invariant rings

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal in $R^G$. Fix $i \geq 0$. If $R^G$ is Gorenstein then, \begin{enumerate} \item $injdim_{R^G} H^i_I(R^G) \leq \dim \ Supp \ H^i_I(R^G). \item H^j_{\mathfrak{m}}(H^i_I(R^G))$is injective, where$\mathfrak{m}$is any maximal ideal of$R^G. \item \mu_j(P, H^i_I(R^G)) = \mu_j(P^\prime, H^i_{IR}(R))$where$P^\prime$is any prime in$R$lying above$P. \end{enumerate} We also prove that if P$is a prime ideal in$R^G$with$R^G_P$\textit{not Gorenstein} then either the bass numbers$\mu_j(P, H^i_I(R^G)) $is zero for all$j$or there exists$c$such that$\mu_j(P, H^i_I(R^G)) = 0 $for$j < c$and$\mu_j(P, H^i_I(R^G)) > 0$for all$j \geq c$.Comment: Some of the results in the previous version was already known by work of N\'{u}\~{n}ez-Betancourt. Those results have been removed in this version. Also some new results are adde

Rings of invariants of finite groups when the bad primes exist

Let R be a ring (not necessarily with 1) and G be a finite group of automorphisms of R. The set B(R, G) of primes p such that p | |G| and R is not p-torsion free, is called the set of bad primes. When the ring is |G|-torsion free, i.e., B(R, G) is empty set, the properties of the rings R and R^G are closely connected. The aim of the paper is to show that this is also true when B(R, G) is not empty set under natural conditions on bad primes. In particular, it is shown that the Jacobson radical (resp., the prime radical) of the ring R^G is equal to the intersection of the Jacobson radical (resp., the prime radical) of R with R^G; if the ring R is semiprime then so is R^G; if the trace of the ring R is nilpotent then the ring itself is nilpotent; if R is a semiprime ring then R is left Goldie iff the ring R^G is so, and in this case, the ring of G-invariants of the left quotient ring of R is isomorphic to the left quotient ring of R^G and im (R^G)\leq im (R)\leq |G| im (R^G)

Gravastars in f(R,G)f(R,G) Gravity

In this paper, we discuss some feasible features of gravastar that were firstly demonstrated by Mazur and Mottola. It is already established that gravastar associates the de-Sitter spacetime in its inner sector with the Schwarzschild geometry at its exterior through the thin shell possessing the ultra-relativistic matter. We have explored the singularity free spherical model with a particular equation of state under the influence of $f(R,G)$ gravity, where $R$ is the Ricci scalar and $G$ is the Gauss-Bonnet term. The interior geometry is matched with a suitable exterior using Israel formalism. Also, we discussed a feasible solution of gravastar which describes the other physically sustainable factors under the influence of $f(R,G)$ gravity. Different realistic characteristics of the gravastar model are discussed, in particular, shell's length, entropy, and energy. A significant role of this particular gravity is examined for the sustainability of gravastar model.Comment: 21 pages, 4 figure

On a Conjecture of Randi\'{c} Index and Graph Radius

The Randi\'{c} index $R(G)$ of a graph $G$ is defined as the sum of $(d_i d_j)^{-1/2}$ over all edges $v_i v_j$ of $G$, where $d_i$ is the degree of the vertex $v_i$ in $G$. The radius $r(G)$ of a graph $G$ is the minimum graph eccentricity of any graph vertex in $G$. Fajtlowicz(1988) conjectures $R(G) \ge r(G)-1$ for all connected graph $G$. A stronger version, $R(G) \ge r(G)$, is conjectured by Caporossi and Hansen(2000) for all connected graphs except even paths. In this paper, we make use of Harmonic index $H(G)$, which is defined as the sum of $\frac{2}{d_i+d_j}$ over all edges $v_i v_j$ of $G$, to show that $R(G) \ge r(G)-31/105(k-1)$ for any graph with cyclomatic number $k\ge 1$, and $R(T)> r(T)+1/15$ for any tree except even paths. These results improve and strengthen the known results on these conjectures

Energy conditions in f(R,G)f(R,G) gravity

Modified gravity is one of the most favorable candidates for explaining the current accelerating expansion of the Universe. In this regard, we study the viability of an alternative gravitational theory, namely $f(R,G)$, by imposing energy conditions. We consider two forms of $f(R,G)$, commonly discussed in the literature, which account for the stability of cosmological solutions. We construct the inequalities obtained by energy conditions and specifically apply the weak energy condition using the recent estimated values of the Hubble, deceleration, jerk and snap parameters to probe the viability of the above-mentioned forms of $f(R, G)$.Comment: 10 pages, 6 figures. arXiv admin note: text overlap with arXiv:1011.4159, arXiv:1012.0953 by other author

Restricted size Ramsey number for P3P_3 versus cycles

Let $F$, $G$ and $H$ be simple graphs. We say $F \rightarrow (G, H)$ if for every $2$-coloring of the edges of $F$ there exists a monochromatic $G$ or $H$ in $F$. The Ramsey number $r(G, H)$ is defined as $r(G, H) = min\{|V (F)|: F \rightarrow (G, H)\}$, while the restricted size Ramsey number $r^{*}(G, H)$ is defined as $r^{*}(G, H) = min\{|E (F)|: F \rightarrow (G, H) , |V (F) | = r(G, H)\}$. In this paper we determine previously unknown restricted size Ramsey numbers $r^{*}(P_3, C_n)$ for $7 \leq n \leq 12$. We also give new upper bound $r^{*}(P_3, C_n) \leq 2n-2$ for even $n \geq 8$

Randi\'c index, diameter and the average distance

The Randi\'c index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randi\'c index $R(G)$ with relations to the diameter $D(G)$ and the average distance $\mu(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $\delta(G)$, if $\delta(G)\geq 5$, then $R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$; if $\delta(G)\geq n/5$ and $n\geq 15$, $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq \mu(G)$. Furthermore, for any arbitrary real number $\varepsilon \ (0<\varepsilon<1)$, if$\delta(G)\geq \varepsilon n$, then$\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$and$R(G)\geq \mu(G)$hold for sufficiently large$n$.Comment: 7 page

Reliability Polynomials and their Asymptotic Limits for Families of Graphs

We present exact calculations of reliability polynomials $R(G,p)$ for lattice strips $G$ of fixed widths $L_y \le 4$ and arbitrarily great length $L_x$ with various boundary conditions. We introduce the notion of a reliability per vertex, $r(\{G\},p) = \lim_{|V| \to \infty} R(G,p)^{1/|V|}$ where $|V|$ denotes the number of vertices in $G$ and $\{G\}$ denotes the formal limit $\lim_{|V| \to \infty} G$. We calculate this exactly for various families of graphs. We also study the zeros of $R(G,p)$ in the complex $p$ plane and determine exactly the asymptotic accumulation set of these zeros ${\cal B}$, across which $r(\{G\})$ is nonanalytic.Comment: 56 pages, latex, 16 figures, version to appear in J. Stat. Phy

Constructions of f(R,G,T)f(R,G,\mathcal{T}) Gravity from Some Expansions of the Universe

Here we propose the extended modified gravity theory named as $f(R,G,\mathcal{T})$ gravity where $R$ is the Ricci scalar, $G$ is the Gauss-Bonnet invariant and $\mathcal{T}$ is the trace of the stress-energy tensor. We derive the gravitational field equations in $f(R,G,\mathcal{T})$ gravity by taking least action principle. Next we construct the $f(R,G,\mathcal{T})$ in terms of $R$, $G$ and $\mathcal{T}$ in de Sitter as well as power law expansion. We also construct $f(R,G,\mathcal{T})$ if the expansion follows the finite time future singulary (big rip singularity). We investigate the energy conditions in this modified theory of gravity and examine the validities of all energy conditions. Finally, we analyze the stability of the constructed modified gravity.Comment: 19 pages, 18 figure

On the Roman bondage number of a graph

A Roman dominating function on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,2\}$ such that every vertex $v\in V$ with $f(v)=0$ has at least one neighbor $u\in V$ with $f(u)=2$. The weight of a Roman dominating function is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the Roman domination number, denoted by $\gamma_{R}(G)$. The Roman bondage number $b_{R}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $E'\subseteq E(G)$ for which $\gamma_{R}(G-E')>\gamma_R(G)$. In this paper, we first show that the decision problem for determining $b_{\rm R}(G)$ is NP-hard even for bipartite graphs and then we establish some sharp bounds for $b_{\rm R}(G)$ and characterizes all graphs attaining some of these bounds.Comment: 15 pages, 35 reference