In this note it is shown that the number of cycles of a linear hypergraph is bounded below by its cyclomatic number.

The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index. In this paper, we study the \(PI\) index of thorn graphs, and we present a generally useful method which can reduce the computational amount of \(PI\) index strikingly.

The concept of the sum graph and integral sum graph were introduced by F. Harary. In this paper, we gain some upper and lower bounds on the sum number and the integral sum number of a graph and these bounds are sharp, and some new properties on the integral sum graph. Using these results, we could directly investigate and determine the exclusive integral sum numbers, the exclusive sum numbers, the sum numbers and the integral sum numbers of the graphs \(K_n\backslash E(2P_3)\), \(K_n\backslash E(P_3)\) and any graph \(H\) with minimum degree \(\delta(H) = n-2\) respectively as \(2\) is more than a given number. Then they will be the beginning of a new thought of research on the (exclusive) sum graph and the (exclusive) integral sum graph.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(g\) and \(f\) be two integer-valued functions defined on \(V(G)\) such that \(0 \leq g(x) \leq f(x)\) for each \(x \in V(G)\). A \((g, f)\)-factor of \(G\) is a spanning subgraph \(F\) of \(G\) such that \(g(x) \leq d_F(x) \leq f(x)\) for each \(x \in V(F)\). A \((g, f)\)-factorization of \(G\) is a partition of \(E(G)\) into edge-disjoint \((g, f)\)-factors. Let \({F} = \{F_1, F_2, \ldots, F_m\}\) be a factorization of \(G\) and \(H\) be a subgraph of \(G\) with \(m\) edges. If \(F_i\), \(1 \leq i \leq m\), has exactly one edge in common with \(H\), we say that \({F}\) is orthogonal to \(H\). In this paper, it is proved that every \((mg+k-1, mf-k+1)\)-graph contains a subgraph \( {R}\) such that \( {R}\) has a \((g, f)\)-factorization orthogonal to any given subgraph with \(k\) edges of \(G\) if \(f(x) > g(x) \geq 0\) for each \(x \in V(G)\) and \(1 \leq k \leq m\), where \(m\) and \(k\) are two positive integers.

Integral circulant graphs have been proposed as potential candidates for modelling quantum spin networks with perfect state transfer between antipodal sites in the network. We show that the diameter of these graphs is at most \(O(\ln \ln n)\), and further improve the recent result of Saxena, Severini, and Shparlinski.

We present a bijective proof of the hook length formula for rooted trees based on the ideas of the bijective proof of the hook length formula for standard tableaux by Novelli, Pak, and Stoyanovskii \([10]\). In Section \(4\), we present another bijection for the formula.

In this paper, we give sufficient conditions for the existence of kernels by monochromatic directed paths (m.d.p.) in digraphs with quasi-transitive colorings. Let \(D\) be an \(m\)-colored digraph. We prove that if every chromatic class of \(D\) is quasi-transitive, every cycle is quasi-transitive in the rim and \(D\) does not contain polychromatic triangles, then \(D\) has a kernel by m.d.p. The same result is valid if we preserve the first two conditions before and replace the last one by: there exists \(k \geq 4\) such that every \(\overrightarrow{C}_k\) is quasi-monochromatic and every \(\overrightarrow{C}_{k-1}\) (\(3 \leq l \leq k-1\)) is not polychromatic. Finally, we also show that if every chromatic class of \(D\) is quasi-transitive, every cycle in \(D\) induces a quasi-transitive digraph and \(D\) does not contain polychromatic \(\overrightarrow{C}_3\), then \(D\) has a kernel by m.d.p. Some corollaries are obtained for the existence of kernels by m.d.p. in \(m\)-colored tournaments.

The determination of the zero-capacity of a noisy channel has inspired research on the independence number of the strong product of odd cycles. The independence number for two infinite families of the strong product of three odd cycles is considered in this paper. In particular, we present the independence number of \(C_7 \boxtimes C_9 \boxtimes C_{2k+1}\) and an upper bound on the independence number of \(C_{13} \boxtimes C_3 \boxtimes C_{2k+1}\). The results are partially obtained by a computer search.

The strong product \(G_1 \boxtimes G_2\) of graphs \(G_1\) and \(G_2\) is the graph with \(V(G_1) \times V(G_2)\) as the vertex set, and two distinct vertices \((x_1,x_2)\) and \((y_1,y_2)\) are adjacent whenever for each \(i \in \{1,2\}\) either \(x_i = y_i\) or \(x_iy_i \in E(G_i)\).

An edge irregular total \(k\)-labeling \(\varphi: V \cup E \to \{1,2,\ldots,k\}\) of a graph \(G = (V, E)\) is a labeling of vertices and edges of \(G\) in such a way that for any different edges \(xy\) and \(x’y’\) their weights \(\varphi(x) + \varphi(xy) + \varphi(y)\) and \(\varphi(x’) + \varphi(x’y’) + \varphi(y’)\) are distinct. The total edge irregularity strength, \(\text{tes}(G)\), is defined as the minimum \(k\) for which \(G\) has an edge irregular total \(k\)-labeling.

We have determined the exact value of the total edge irregularity strength of the strong product of two paths \(P_n\) and \(P_m\).

Given \(m, n\) and \(2 \leq l \leq mn\), we study the problem of separating \(l\) symbols on an \(m \times n\) array such that the minimum \(\ell_1\) distance between any two of the \(l\) symbols is as large as possible. This problem is similar in nature to the well-known Tammes’ problem where one tries to achieve the largest angular separation for a given number of points on a \(2-D\) or higher dimensional sphere. It is also closely related to the well-studied problem of constructing optimal interleaving schemes for correcting error bursts in multi-dimensional digital data where a burst can be an arbitrarily shaped connected region in the array. Moreover, the interest in studying this problem also arises from considerations of minimizing the risk of multiple nearby node failures in a distributed data storage system (or a similar industrial network) in the event of a relatively large scale random disruption. We derive bounds on the maximum possible distance of separation for general \(m,n\) and \(l\), and provide also optimal constructions in several special cases including small and large \(l\) values, small \(m\) (or \(n\)) values, and \(n-1 \geq (l-1)(m-1)\).

In this paper, we apply the concepts of intuitionistic fuzzy sets to coalgebras. We give the definition of intuitionistic fuzzy subcoalgebras and investigate some properties of intuitionistic fuzzy subcoalgebras. Considering the applications of intuitionistic fuzzy subcoalgebras, we discuss their properties under homomorphisms of coalgebras.

In this paper, the joint tree method of graph embeddings, which was introduced by Liu, is generalized to digraph embeddings. The genus distributions of a new type of digraphs in orientable surfaces are determined.

In this paper, the \(m\)-hull sets in the join and composition of two connected graphs are characterized and their \(m\)-hull numbers are shown to be direct consequences of these characterizations.

The generalized de Bruijn digraph denoted by \(G_B(n,m)\) is the digraph \((V, A)\) where \(V = \{0,1,\ldots,m-1\}\) and \((i,j) \in A\) if and only if \(j \equiv ni + \alpha \pmod{m}\) for some \(\alpha \in \{0,1,\ldots,n-1\}\). By replacing each arc of \(G_B(n,m)\) with an undirected edge and eliminating loops and multi-edges, we obtain a generalized undirected de Bruijn graph \(UG_B(n,m)\). In this paper, we prove that the diameter of \(UG_B(n,m)\) is equal to 3 whenever \(n \geq 2\) and \(n^2 + (\frac{\sqrt{5}+1}{2})\leq m \leq 2n^2.\)

The zeroth-order general Randić index of a graph \(G\) is defined as \({}^{0}{}{R}_\alpha = \sum\limits_{v\in V(G)} d(v)^\alpha\)
where \(d(v)\) is the degree of the vertex \(v\) in \(G\) and \(\alpha\) is an arbitrary real number. In the paper, we give sharp lower and upper bounds on the zeroth-order general Randić index of cacti.

Let \(G\) be a connected \(k\)-colourable graph of order \(n \geq k\). A subgraph \(H\) of \(G\) is \(k\)-colourfully panconnected in \(G\) if there is a \(k\)-colouring of \(G\) such that the colours are close together in \(H\), in two different senses (called variegated and panconnected) to be made precise. Let \(s_k(G)\) denote the smallest number of edges in a spanning \(k\)-colourfully panconnected subgraph \(H\) of \(G\). It is conjectured that \(s_k(G) = n-1\) if \(k \geq 4\) and \(G\) is not a circuit (a connected \(2\)-regular graph) with length \(\equiv 1 \pmod{k}\). It is proved that \(s_k(G) = n-1\) if \(G\) contains no circuit with length \(\equiv 1 \pmod{k}\), and \(s_k(G) \leq 2n-k-1\) whenever \(k \geq 4\).

Multisender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify authenticity of the received message. In this paper, we give the model of multisender authentication codes and the calculation formulas on probability of success in attacks by malicious groups of senders. A construction of multisender authentication codes from symplectic geometry over finite fields is given, and the parameters and the probabilities of deceptions are also calculated.

Let \((X,{B})\) be an \(\alpha\)-fold block design with block size \(4\). If a star is removed from each block of \({B}\), the resulting collection of triangles \({T}\) is a partial \(\lambda\)-fold triple system \((X,{T})\). If the edges belonging to the deleted stars can be arranged into a collection of triangles \({S}^*\), then \((X,{T} \cup {S}^*)\) is an \(\lambda\)-fold triple system, called a metamorphosis of the \(\lambda\)-fold block design \((X, {B})\) into a \(4\)-fold triple system.

Label the elements of each block \(b\) with \(b_1, b_2, b_3\) and \(b_4\) (in any manner). For each \(i = 1,2,3,4\), define a set of triangles \({T}_i\) and a set of stars \({S}_i\) as follows: for each block \(b = (b_1, b_2, b_3, b_4)\) belonging to \({B}\), partition \(b\) into a triangle and a star centered at \(b_i\), and place the triangle in \({T}_i\) and the star in \({S}_i\). Then \((X,\mathcal{T}_i)\) is a partial \(\alpha\)-fold triple system.

Now if the edges belonging to the stars in \({S}_i\) can be arranged into a collection of triangles \({S}_i^*\), then \((X,{T}_i \cup {S}_i^*)\) is an \(\lambda\)-fold triple system and we say that \(M_i = (X,{T}_i \cup {S}_i^*)\) is the \(i\)th metamorphosis of \((X,{B})\).

The full metamorphosis of \((X,{B})\) is the set of four metamorphoses \(\{M_1, M_2, M_3, M_4\}\). The purpose of this work is to give a complete solution of the following problem: For which \(n\) and \(\lambda\) does there exist an \(\lambda\)-fold block design with block size \(4\) having a full metamorphosis into \(\lambda\)-fold triple systems?

A labeling of a graph is any map that carries some set of graph elements to numbers (usually to the positive integers). An \((a, d)\)-edge-antimagic total labeling on a graph with \(p\) vertices and \(q\) edges is defined as a one-to-one map taking the vertices and edges onto the integers \(1,2,…,p+q\) with the property that the sums of the labels on the edges and the labels of their endpoints form an arithmetic sequence starting from \(a\) and having a common difference \(d\). Such a labeling is called super if the smallest possible labels appear on the vertices.

We use the connection between \(a\)-labelings and edge-antimagic labelings for determining a super \((a,d)\)-edge-antimagic total labelings of disconnected graphs.

Let \(P\) be an \(n \times n\) array of symbols. \(P\) is called avoidable if for every set of \(z\) symbols, there is an \(n \times n\) Latin square \(L\) on these symbols so that corresponding cells in \(P\) and \(L\) differ. Due to recent work of Cavenagh and Ohman, we now know that all \(n \times n\) partial Latin squares are avoidable for \(n \geq 4\). Cavenagh and Ohman have shown that partial Latin squares of order \(4m + 1\) for \(m \geq 1\) [1] and \(4m – 1\) for \(m \geq 2\) [2] are avoidable. We give a short argument that includes all partial Latin squares of these orders of at least \(9\). We then ask the following question: given an \(n \times n\) partial Latin square \(P\) with some specified structure, is there an \(n \times n\) Latin square \(L\) of the same structure for which \(L\) avoids \(P\)? We answer this question in the context of generalized sudoku squares.

For a graph \(G\) with vertices labeled \(1,2,\ldots,n\) and a permutation \(\alpha\) in \(S_n\), the symmetric group on \(\{1,2,\ldots,n\}\), the \(\alpha\)-generalized prism over \(G\), \(\alpha(G)\), consists of two copies of \(G\), say \(G_x\) and \(G_y\), along with the edges \((x_i, y_{\alpha(i)})\), for \(1 \leq i \leq n\). In [10], the importance of building large graphs by using generalized prisms is indicated. A graph \(G\) is supereulerian if it has a spanning eulerian subgraph. In this note, we consider results of the form that if \(G\) has property \(P\), then for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. As a result, we obtain a few properties of \(G\) which implies that for any \(\alpha \in S_{|V(G)|}\), \(\alpha(G)\) is supereulerian. Also, while the permutations are restricted, the related result is discussed.

Using the model of words, we give bijective proofs of Gould-Mohanty’s and Raney-Mohanty’s identities, which are respectively multivariable generalizations of Gould’s identity

\[\sum\limits_{k=0}^{n} \left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
= \sum\limits_{k=0}^{n}
\left(
\begin{array}{c}
x+\epsilon-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y-\epsilon+kz \\
n-k \\
\end{array}
\right)
\]

and Rothe’s identity
\[\sum\limits_{k=0}^{n}\frac{x}{x-kz}
\left(
\begin{array}{c}
x-kz \\
k \\
\end{array}
\right)
\left(
\begin{array}{c}
y+kz \\
n-k \\
\end{array}
\right)
=
\left(
\begin{array}{c}
x+y \\
n \\
\end{array}
\right)\]

Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Broušek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure is the line graph of a multigraph.

For a graph \(G = (V,E)\), a function \(f : V \rightarrow \{0,1,2\}\) is called a Roman dominating function (RDF) if for any vertex \(v\) with \(f(v) = 0\), there is at least one vertex \(w\) in its neighborhood with \(f(w) = 2\).

The weight of an RDF \(f\) of \(G\) is the value \(f(V) = \sum_{v\in V} f(v)\). The minimum weight of an RDF of \(G\) is its Roman domination number, denoted by \(\gamma_R(G)\). In this paper, we show that \(\gamma_R(G) + 1 \leq \gamma_R(\mu(G)) \leq \gamma_R(G) + 2\), where \(\mu(G)\) is the Mycielekian graph of \(G\), and then characterize the graphs achieving equality in these bounds.

A graph is said to be cordial if it has a \(0-1\) labeling that satisfies certain properties. A fan \(F_n\) is the graph obtained from the join of the path \(P_n\) and the null graph \(N_1\). In this paper, we investigate the cordiality of the join and the union of pairs of fans and graphs consisting of a fan with a path, and a cycle.

We consider the problem of covering a unit cube with smaller cubes. The size of a cube is given by its side length and the size of a covering is the total size of the cubes used to cover the unit cube. We denote by \(g_3(n)\) the smallest size of a minimal covering using \(n\) cubes. We present tight results for the upper and lower bounds of \(g_3(n)\).

Let \(G\) be a graph. The cardinality of any largest independent set of vertices in \(G\) is called the independence number of \(G\) and is denoted by \(\alpha(G)\). Let \(a\) and \(b\) be integers with \(0 \leq a \leq b\). If \(a = b\), it is assumed that \(G\) be a connected graph, furthermore, \(a \geq \alpha(G)\), \(a/|V(G)| = 0 \pmod{2}\) if \(a\) is odd. We prove that every graph \(G\) has an \([a, b]\)-factor if its minimum degree is at least \((\frac{b+\alpha(G)a-\alpha(G)}{b})\lfloor \frac{b+\alpha(G)a}{2\alpha(G)} \rfloor -\frac{\alpha(G)}{b}(\lfloor \frac{b+\alpha(G)a}{2\alpha(G)}\rfloor )^2+ \theta\frac{\alpha(G)^2}{b}+\frac{a}{b}\alpha(G)\), where \(\theta = 0\) if \(a < b\), and \(\theta = 1\) if \(a = b\). This degree condition is sharp.

Suppose that graphs \(H\) and \(G\) are graceful, and that at least one of \(H\) and \(G\) has an \(\alpha\)-labeling. Four graph operations on \(H\) and \(G\) are provided. By utilizing repeatedly or in turn the four graph operations, we can construct a large number of graceful graphs. In particular, if both \(H\) and \(G\) have \(\alpha\)-labelings, then each of the graphs obtained by the four graph operations on \(H\) and \(G\) has an \(\alpha\)-labeling.

In this paper, we present three algebraic constructions of authentication codes from power functions over finite fields with secrecy and realize an application of some properties about authentication codes in [1]. The first and the third class are optimal. Some of the codes in the second class are optimal, and others in the second class are asymptotically optimal. All authentication codes in the three classes provide perfect secrecy.

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by \(q\)-series and compositions exhibiting particular patterns are specified by generating functions for these patterns. Here we view compositions as alternating sequences of partitions (i.e., alternating blocks) and obtain results for the asymptotic expectations of the number of such blocks (or parts per block) for different ways of defining the blocks.

For any integer \(k \geq 1\), a signed (total) \(k\)-dominating function is a function \(f : V(G) \rightarrow \{-1, 1\}\) satisfying \(\sum_{u \in N(v)} f(u) > k\) (\(\sum_{w \in N[v]} f(w) \geq k\)) for every \(v \in V(G)\), where \(N(v) = \{u \in V(G) | uv \in E(G)\}\) and \(N[v] = N(v) \cup \{v\}\). The minimum of the values of \(\sum_{v \in V(G)} f(v)\) , taken over all signed (total) \(k\)-dominating functions \(f\), is called the signed (total) \(k\)-domination number and is denoted by \(\gamma_{kS}(G)\) (\(\gamma’_{kS}(G)\), resp.). In this paper, several sharp lower bounds of these numbers for general graphs are presented.

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.

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.