The main aim of this paper is to present the idea of \(L\)-presheaves on a topological space \(X\). Categorical properties of \(L\)-presheaves are studied. The nature of \(L\)-presheaves locally in the neighbourhood of some point is summarized. This aim required constructing the notions of category of \(L\)-sets, \(L\)-direct systems and their \(L\)-limits and \(L\)-functors with their \(L\)-natural transformations. We prove that the ”\(L\)-stalk” is an \(L\)-functor from the category of \(L\)-presheaves to the category of \(L\)-sets.

A \( t \)-strong biclique covering of a graph \( G \) is an edge covering \(
E(G) = \bigcup_{i=1}^{t} E(H_i)\) where each \( H_i \) is a set of disjoint bicliques; say \( H_{i,1}, …, H_{i,r_i} \), such that the graph \( G \) has no edge between \( H_{i,k} \) and \( H_{i,j} \) for any \( 1 \leq j < k \leq r_i \). The strong biclique covering index \( S(G) \) is the minimum number \( t \) for which there exists a \( t \)-strong biclique covering of \( G \). In this paper, we study the strong biclique covering index of graphs. The strong biclique covering index of graphs was introduced in [H. Hajiabolhassan, A. Cheraghi, Bounds for Visual Cryptography Scheme, Discrete Applied Mathematics, 158 (2010), 659-665] to study the pixel expansion of visual cryptology. We present a lower bound for the strong biclique covering index of graphs and also we introduce upper bounds for different products of graphs.

We introduce vertex-transitive graphs \(\Gamma_n\), that are also embeddings of the strong product of triangular graphs \(L(K_n)\) and the complete graph \(K_2\). For any prime \(p\), linear codes obtained from the row span of incidence matrices of the graphs over \(\mathbb{F}_p\), are considered; their main parameters (length, dimension and minimum distance) and automorphism groups are determined. Unlike most codes that have been obtained from incidence and adjacency matrices of regular graphs by others, binary codes from the row span of incidence matrices of \(\Gamma_n\) have other minimum words apart from the rows of the matrices. Using a specific information set, PD-sets for full permutation decoding of the codes are exhibited.

Let \(G\) be a connected graph of order \(p \geq 2\). The closed interval \(I[x,y]\) consists of all vertices lying on some \(x-y\) geodesic of \(G\). If \(S\) is a set of vertices of \(G\), then \(I[S]\) is the union of all sets \(I\{x, y\}\) for \(x, y \in S\). The geodetic number \(g(G)\) is the minimum cardinality among the subsets \(S\) of \(V(G)\) with \(I[S] = V\). A geodetic set of cardinality \(g(G)\) is called a \(g\)-set of \(G\). For any vertex \(z\) in \(G\), a set \(S_x \subseteq V\) is an \(x\)-geodominating set of \(G\) if each vertex \(v \in V\) lies on an \(z-y\) geodesic for some element \(y\) in \(S_z\). The minimum cardinality of an \(x\)-geodominating set of \(G\) is defined as the \(x\)-geodomination number of \(G\), denoted by \(g_x(G)\) or simply \(g_x\). An \(x\)-geodominating set \(S_x\) of cardinality \(g_x(G)\) is called a \(g_x\)-set of \(G\). If \(S_x \cup \{x\}\) is a \(g\)-set of \(G\), then \(x\) is called a geo-vertex of \(G\). The set of all geo-vertices of \(G\) is called the geo-set of \(G\) and the number of geo-vertices of \(G\) is called the geo-number of \(G\) and it is denoted by \(gn(G)\). For positive integers \(r, d\) and \(n \geq 2\) with \(r < d \leq 2r\), there exists a connected graph \(G\) of radius \(r\), diameter \(d\) and \(gn(G) = n\). Also, for each triple \(p, d\) and \(n\) with \(3 \leq d \leq p – 1, 2 \leq n \leq p – 2\) and \(p – d – n + 1 \geq 0\), there exists a graph \(G\) of order \(p\), diameter \(d\) and \(gn(G) = n\). If the \(x\)-geodomination number \(g_x(G)\) is same for every vertex \(x\) in \(G\), then \(G\) is called a vertex geodomination regular graph or for short VGR-graph. If \(S \cup \{x\}\) is same for every vertex \(x\) in \(G\), then \(G\) is called a perfect vertex geodomination graph or for short PVG-graph. We characterize a PVG-graph.

The Wiener index, one of the oldest molecular topological descriptors used in mathematical chemistry, was well-studied during the past decades. For a graph \(G\), its Wiener index is defined as \(W(G) = \sum\limits_{\{u, v\} \subseteq V(G)} d_G(u, v)\), where \(d_G(u, v)\) is the distance between two vertices \(u\) and \(v\) in \(G\). In this paper, we study the Wiener index of a class of composite graph, namely, double graph. We reveal the relation between the Wiener index of a given graph and the one of its double graph as well as the relation between Wiener index of a given graph and the one of its \(k\)-iterated double graph. As a consequence, we determine the graphs with the maximum and minimum Wiener index among all double graphs and \(k\)-iterated double graphs of connected graphs of the same order, respectively.

The set of unicyclic graphs with \(n\) vertices and diameter \(d\) is denoted by \(\mathcal{U}_{n,d}\). For \(3 \leq i \leq d\), let \(P_{n-d-1}(i)\) be the graph obtained from path \(P_{d+1}: v_1 v_2 \ldots v_{d+1}\) by adding \(n-d-1\) pendant edges at \(v_i\), and \(U_{n-d-2}(i)\) be the graph obtained from \(P_{n-d-1}(i)\) by joining \(v_{i-2}\) and a pendant neighbor of \(v_{i}\). In this paper, we determine all unicyclic graphs in \(\mathcal{U}_{n,d}\) whose largest Laplacian eigenvalue is greater than \(n-d+2\). For \(n-d \geq 6\) and \(G \in \mathcal{U}_{n,d}\), we prove further that the largest Laplacian eigenvalue \(\mu(G) \leq \max\{\lambda(U_{n,d-2}(i)) \mid 3 \leq i \leq d\}\), and conjecture that \(\mathcal{U}_{n,d}.\) is the unique graph which has the greatest value of the greatest Laplacian eigenvalue in \(\mathcal{U}_{n,d}\). We also prove that the conjecture is true for \(3 \leq d \leq 6\).

The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI index with respect to the extremal simple pericondensed hexagonal systems and we solve it completely.

Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices. Let \(G\) be a finite simple graph. A \(G\)-design (\(G-GD_\lambda)(v)\) (\(G\)-packing (\(G-PD_\lambda)(v)\), \(G\)-covering (\(G-CD_\lambda)(v)\)) of \(K_v\) is a pair \((X, \mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined exactly (at most, at least) in \(\lambda\) blocks. In this paper, we will discuss the maximum packing designs and the minimum covering designs for four particular graphs each with six vertices and nine edges.

Let \(a\) and \(b\) be integers such that \(1 \leq a < b\), and let \(G\) be a graph of order \(n\) with \(n \geq \frac{(a+b)(2a+2b-3)}{a+1}\) and the minimum degree \(\delta(G) \geq \frac{(b-1)^2-(a+1)(b-a-2)}{a+1} \). Let \(g(x)\) and \(f(x)\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\) for each \(x \in V(G)\). We prove that if \(|N_G(x) \cup N_G(y)| \geq \frac{(b-1)n}{a+b} \) for any two nonadjacent vertices \(x\) and \(y\) in \(G\), then \(G\) has a \((g, f)\)-factor. Furthermore, it is shown that the result in this paper is best possible in some sense.

Let \(D\) be a digraph with order at least two. The transformation digraph \(D^{++-}\) is the digraph with vertex set \(V(D) \cup A(D)\) in which \((x, y)\) is an arc of \(D^{++-}\) if one of the following conditions holds:(i) \(x, y \in V(D)\), and \((x, y)\) is an arc of \(D\);(ii) \(x, y \in A(D)\), and the head of \(x\) is the tail of \(y\);(iii) \(x \in V(D), y \in A(D)\), and \(x\) is not the tail of \(y\);(iv) \(x \in A(D), y \in V(D)\), and \(y\) is not the head of \(x\).In this paper, we determine the regularity and diameter of \(D^{++-}\). Furthermore, we characterize maximally-arc-connected or super-arc-connected \(D^{++-}\). We also give sufficient conditions for this kind of transformation digraph to be maximally-connected or super-connected.