Let $R$ be a commutative ring and $mathbb{A}(R)$ be the set of all ideals with non-zero annihilators. Assume that $mathbb{A}^*(R)=mathbb{A}(R)diagdown {0}$ and $mathbb{F}(R)$ denote the set of all finitely generated ideals of $R$. In this paper, we introduce and investigate the {it finitely generated subgraph} of the annihilating-ideal graph of $R$, denoted by $mathbb{AG}_F(R)$. It is the (undi...

the reverse degree distance of a connected graph $g$ is defined in discrete mathematical chemistry as [ r (g)=2(n-1)md-sum_{uin v(g)}d_g(u)d_g(u), ] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $g$, respectively, $d_g(u)$ is the degree of vertex $u$, $d_g(u)$ is the sum of distance between vertex $u$ and all other vertices of $g$, and $v(g)$ is the ...

We construct a graph of order 384, the smallest known trivalent graph of girth 14. AMS Subject Classifications: 05D25, 05D35 In this note we use a construction technique that can be viewed as a kind of generalized Cayley graph. The vertex set V of such a graph consists of the elements in multiple copies of some finite group G. The action of G on V is determined by the regular action on each of ...

The Moore bound m(d, k) = 1 + d ∑k−1 i=0 (d − 1)i is a lower bound for the number of vertices of a graph by given girth g = 2k + 1 and minimal degree d. Hoffmann and Singleton [5], Bannai and Ito [1], Damerell [4] showed that graphs with d > 2 tight to this bound can only exist for girth 5 and degree 3, 7, 57. The difference to the Moore bound by given girth is called the excess of a graph. In ...

The annihilating-ideal graph of a commutative ring $R$ is denoted by $AG(R)$, whose vertices are all nonzero ideals of $R$ with nonzero annihilators and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=0$. In this article, we completely characterize rings $R$ when $gr(AG(R))neq 3$.

A family of trivalent graphs is described that includes most of the known trivalent cages. A new graph in this family is the smallest trivalent graph of girth 17 yet discovered.

The well-known Moore bound M(k, g) serves as a universal lower bound for the order of k-regular graphs of girth g. The excess e of a k-regular graph G of girth g and order n is the difference between its order n and the corresponding Moore bound, e = n −M(k, g). We find infinite families of parameters (k, g), g > 6 and even, for which we show that the excess of any k-regular graph of girth g is...

We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism ...

This article presents a method for constructing large girth column-weight 2 QC-LDPC codes. A distance graph is first constructed using an existing method. The distance graph is then converted into a Tanner graph. The proposed method could easily construct codes with girths large than 12 and is more flexible compared to previous methods. Obtained codes show good bit error rate performance compar...

For every r ∈ N, let θr denote the graph with two vertices and r parallel edges. The θr-girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to θr. This notion generalizes the usual concept of girth which corresponds to the case r = 2. In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213–225, 2003], Kühn and Osthus showed that gra...