Let G = (V, E) be a graph with p vertices and q edges. A has product-set labeling if there exist an injective function f: V(G) → P (N) such that the induced edge f∗: P(N) is defined as f∗ (μν) f (μ) ∗ (ν) ∀μ, ν ∈ E(G).f {ab: (μ), b }. In this paper we investigate how works for zero-divisor line of graphs.

Let $R$ be an associative ring with identity. A ring $R$ is called reversible if $ab=0$, then $ba=0$ for $a,bin R$. The quasi-zero-divisor graph of $R$, denoted by $Gamma^*(R)$ is an undirected graph with all nonzero zero-divisors of $R$ as vertex set and two distinct vertices $x$ and $y$ are adjacent if and only if there exists $0neq rin R setminus (mathrm{ann}(x) cup mathrm{ann}(y))$ such tha...

is of Hausdorff dimension at most n− p, by a theorem of Federer-Ziemer [3]. And for functions of bounded variation, we have Hn−1 (Σ∗) = 0, see section 5.9 of [1] for more information on fine properties of BV functions. Hence our result concerning Σ, the Lebesgue set of u of the value zero makes sense because s > n− p. In order to show Theorem 1, we will need a Poincaré type inequality which wil...

We present another set of experiments conducted on the widely used IAPRTC-12 [6] dataset. We use the same tag annotation and image training-test split as described in [7] for our experiments. There are 291 unique tags and 19627 images in IAPRTC12. The dataset is split to 17341 training images and 2286 testing images. We further separate 15% from the training images as our validation set. Table ...