A visual cryptoGRAPHy scheme based on a given GRAPH G is a method to distribute a secret image among the vertices of G, the participants, so that a subset of participants can recover the secret image if they contain an edge of G, by stacking their shares, otherwise they can obtain no information regarding the secret image. In this paper a maximal independent sets of the GRAPH G was applied to propose a lower bound on the pixel expansion of visual cryptoGRAPHy schemes with GRAPH ACCESS STRUCTURE G (G). In addition a lower bound on the pixel expansion of basis matrices C5 and Peterson GRAPH ACCESS STRUCTURE were presented.

In this paper, we propose a new method for GRAPH summarization named GSSC, GRAPH Summarization based on both STRUCTURE and Concepts. In this method, an attributed GRAPH is summarized by considering both of its topology and related concepts. In this method, for a given attributed GRAPH a new GRAPH is constructed that an edge in this new GRAPH represents structural and conceptual similarity of its two end points. Structural and conceptual similarity of two nodes not necessarily has the equal amount of importance in the weight of the resulting edge. For example, for a special case such as query answering, STRUCTURE can be more important and vice versa. Similarity of two nodes is computed based on Jaccard similarity. This method has some advantages such as flexibility, simplicity, learning capability, user-orientation that makes it a better method for GRAPH summarization. We implemented our method and the method proposed by Bei and evaluated these two methods on real-life dataset HEP_TH. Our experimental results showed effectiveness and efficiency of our proposed method.

For a set ‎$‎X‎$ ‎and a ‎topological space ‎$‎(X,\tau‎)$, ‎let‎ ‎‎$‎\Gamma_{_X}(\tau)‎$ ‎be a‎ ‎GRAPH ‎with ‎vertex ‎set ‎‎$‎\tau\setminus \{\emptyset,X\}‎$ ‎in ‎which ‎two ‎vertices ‎‎$‎A_1‎$ ‎and ‎‎$‎A_2‎$ ‎are ‎adjacent ‎just ‎when ‎‎$‎A_1\cup A_2=X‎$‎. In this paper and among some other results, we study the maximum and minimum degrees, the matching number, the chromatic number, the chromatic index, ‎the planarity, the Wiener index and the Zagreb index of ‎‎$‎\Gamma_{_X}(\tau)‎$ and we determine their exact values in general cases or in some special topological spaces like ‎$‎T_1‎$‎‎.

In this note, we show that automata theory is a suitable tool for analyzing monopoly-forcing processes. Also, we present the notion of mono-forcing automata by using the monopoly-forcing set for GRAPHs. Moreover, we prove that mono-forcing automata accept more languages than zero-forcing finite automata also, we show that all results in zero-forcing finite automata for complete GRAPHs are established for mono-forcing automata. We examine and deliberate on the language associated with mono-forcing automata for certain specified GRAPHs. Also, we present the style of words that can be recognized with mono-forcing automata. Additionally, we delineate the types of words identifiable by mono-forcing automata. We also describe the configuration of GRAPHs from which mono-forcing automata emerge, generating specific languages. Several examples are provided to elucidate these concepts.