Let \(\lambda K_v\) be the complete multigraph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by \(\lambda\) edges \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-packing design (\(G\)-covering design) of \(K_v\), denoted by \((v,G,\lambda)\)-PD (\((v,G,\lambda)\)-CD) is a pair \((X,B)\), where \(X\) is the vertex set of \(\lambda K_v\) and \(B\) 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 in at most (at least) \(\lambda\) blocks of \(B\). A packing (covering) design is said to be maximum (minimum) if no other such packing (covering) design has more (fewer) blocks. There are four graphs with 7 points, 7 edges and a 5-circle, denoted by \(G_i\), \(i = 1,2,3,4\). In this paper, we have solved the existence problem of the maximum \((v, G_i,\lambda)\)-PD and the minimum \((v, G_i, \lambda)\)-CD.

It was shown by Gaborit et al. [10] that a Euclidean self-dual code over \({GF}(4)\) with the property that there is a codeword whose Lee weight \(\equiv 2 \pmod{4}\) is of interest because of its connection to a binary singly-even self-dual code. Such a self-dual code over \({GF}_4\) is called Type I. The purpose of this paper is to classify all Type I codes of lengths up to 10 and extremal Type I codes of length 12, and to construct many new extremal Type I codes over \({GF}(4)\) of lengths from 14 to 22 and 34. As a byproduct, we construct a new extremal singly-even self-dual binary [36, 18, 8] code, and a new extremal singly-even self-dual binary [68, 34, 12] code with a previously unknown weight enumerator \(W_2\) for \(\beta = 95\) and \(\gamma = 1\).

Let \(j\) and \(k\) be two positive integers. An \(L(j,k)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to the vertices of \(G\) such that the difference between labels of any two adjacent vertices is at least \(j\), and the difference between labels of any two vertices that are at distance two apart is at least \(k\). The minimum range of labels over all \(L(j,k)\)-labelings of a graph \(G\) is called the \(\lambda_{j,k}\)-number of \(G\), denoted by \(\lambda_{j,k}(G)\). Similarly, we can define \(L(j,k)\)-edge-labeling and \(L(j,k)\)-edge-labeling number, \(\lambda’_{j,k}(G)\), of a graph \(G\). In this paper, we show that if \(G\) is \(K_{1,3}\)-free with maximum degree \(\Delta\) then \(\lambda_{j,k}(G) \leq k\lfloor\Delta^2/2\rceil + j\Delta – 1\) except that \(G\) is a 5-cycle and \(j = k\). Consequently, we obtain an upper bound for \(\lambda’_{j,k}(G)\) in terms of the maximum degree of \(L(G)\), where \(L(G)\) is the line graph of \(G\). This improves the upper bounds for \(\lambda’_{2,1}(G)\) and \(\lambda’_{1,1}(G)\) given by Georges and Mauro [Ars Combinatoria \(70 (2004), 109-128]\). As a corollary, we show that Griggs and Yeh’s conjecture that \(\lambda_{2,1}(G) \leq \Delta^2\) holds for all \(K_{1,3}\)-free graphs and hence holds for all line graphs. We also investigate the upper bound for \(\lambda’_{j,k}(G)\) for \(K_{1,3}\)-free graphs \(G\).

Let \(G = (V, E)\) be a hamiltonian graph. A hamiltonian cycle \(C\) of \(G\) is described as \((v_1, v_2, \ldots, v_{n(G)}, v_1)\) to emphasize the order of vertices in \(C\). Thus, \(v_1\) is the beginning vertex and \(v_i\) is the \(i\)-th vertex in \(C\). Two hamiltonian cycles of \(G\) beginning at \(u\), \(C_1 = (u_1, u_2, \ldots, u_{n(G),u_1})\) and \(C_2 = (v_1, v_2, \ldots, v_{n(G)},v_1)\) of \(G\) are independent if \(u_1 = v_1 = u_1\) and \(u_i \neq v_i\) for every \(2 \leq i \leq n(G)\). A set of hamiltonian cycles \(\{C_1, C_2, \ldots, C_k\}\) of \(G\) are mutually independent if they are pairwise independent. The mutually independent hamiltonianicity of graph \(G\), \(\text{IHC}(G)\), is the maximum integer \(k\) such that for any vertex \(u\) there are \(k\)-mutually independent hamiltonian cycles of \(G\) beginning at \(u\). In this paper, we prove that \(\text{IHC}(G) \geq \delta(G)\) for any hamiltonian graph and \(\text{IHC}(G) \geq 2\delta(G) – n(G) + 1\) if \(\delta(G) \geq \frac{n(G)}{2}\). Moreover, we present some graphs that meet the bound mentioned above.

Using connectivity and planarity constraints we characterise all \(5\)-regular planar graphs with diameter \(3\).

In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order \(n\) with girth \(g\) and matching number \(\beta \geq \frac{3g}{2}\), Moreover, we characterize all extremal graphs which attain the lower bound.

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.