We introduce a new technique for constructing pairwise balanced designs and group divisible designs from finite groups. These constructed designs do not yield designs with new parameters, but our construction gives rise to designs having a transitive automorphism group that also preserves the resolution classes.

A shell graph of order \(n\), denoted by \(H(n, n-3)\), is the graph obtained from the cycle \(C_n\) of order \(n\) by adding \(n-3\) chords incident with a common vertex, say \(u\). Let \(v\) be a vertex adjacent to \(u\) in \(C_n\). Sethuraman and Selvaraju \([3]\) conjectured that for all \(k \geq 1\) and for all \(n_i \geq 4\), \(1 \leq i \leq k\), one edge \((uv)\) union of \(k\)-shell graphs \(H(n_i, n_i – 3)\) is cordial. In this paper, we settle this conjecture affirmatively.

In this paper, we give formulas for the sums of generalized order-\(k\) Fibonacci, Pell, and similar other sequences, which we obtain using matrix methods. As applications, we give explicit formulas for the Tribonacci and Tetranacci numbers.

A \((g, f)\)-coloring is a generalized edge-coloring in which each color appears at each vertex \(v\) at least \(g(v)\) and at most \(f(v)\) times, where \(g(v)\) and \(f(v)\) are nonnegative and positive integers assigned to each vertex \(v\), respectively. The minimum number of colors used by a \((g, f)\)-coloring of \(G\) is called the \((g, f)\)-chromatic index of \(G\). The maximum number of colors used by a \((g, f)\)-coloring of \(G\) is called the upper \((g, f)\)-chromatic index of \(G\). In this paper, we determine the \((g, f)\)-chromatic index and the upper \((g, f)\)-chromatic index in some cases.

The Szeged index extends the Wiener index for cyclic graphs by counting the number of atoms on both sides of each bond and summing these counts. This index was introduced by Ivan Gutman at the Attila Jozsef University in Szeged in \(1994\), and is thus called the Szeged index. In this paper, we introduce a novel method for enumerating by cuts. Using this method, an exact formula for the Szeged index of a zig-zag polyhex nanotube \(T = TUHC_6{[p,q]}\) is computed for the first time.

In this study, we showed that an \((n+1)\)-regular linear space, which is the complement of a linear space having points not on \(m+1\) lines such that no three are concurrent in a projective subplane of odd order \(m\), \(m \geq 9\), could be embedded into a projective plane of order \(n\) as the complement of Ostrom’s hyperbolic plane.

For general graphs \(G\), it is known \([6]\) that the minimal length of an addressing scheme, denoted by \(N(G)\), is less than or equal to \(|G| – 1\). In this paper, we prove that for almost all complete bipartite graphs \(K_{m,n}\), \(N(K_{m,n}) = |K_{m,n}| – 2\).

A vertex subversion strategy of a graph \(G\) is a set of vertices \(X \subseteq V(G)\) whose closed neighborhood is deleted from \(G\). The survival subgraph is denoted by \(G/X\). The vertex-neighbor-integrity of \(G\) is defined to be \(VNI(G) = \min\{|X| + r(G/X) : X \subseteq V(G)\},\) where \(r(G/X)\) is the order of a largest component in \(G/X\). This graph parameter was introduced by Cozzens and Wu to measure the vulnerability of spy networks. It was proved by Gambrell that the decision problem of computing the vertex-neighbor-integrity of a graph is NP-complete. In this paper, we evaluate the vertex-neighbor-integrity of the composition graph of two paths.

In this paper, we prove that a matroid with at least two elements is connected if and only if it can be obtained from a loop by a nonempty sequence of non-trivial single-element extensions and series extensions.

Let \(G\) and \(H\) be graphs with a common vertex set \(V\), such that \(G – i \cong H – i\)for all \(i \in V\). Let \(p_i\) be the permutation of \(V – i\) that maps \(G – i\) to \(H – i\), and let \(q_i\) denote the permutation obtained from \(p_i\) by mapping \(i\) to \(i\). It is shown that certain algebraic relations involving the edges of \(G\) and the permutations \(q_iq_j^{-1}\) and \(q_iq_k^{-1}\), where \(i, j, k \in V\) are distinct vertices, often force \(G\) and \(H\) to be isomorphic.

The factorization of matrix \(A\) with entries \(a_{i,j}\) determined by \(a_{i,j} = \alpha a_{i-1,j-1} + \beta a_{i,j-1}\) is derived as \(A = TP^T\). An interesting factorization of matrix \(B\) with entries \(b_{i,j} = \alpha b_{i-1,j} + \beta b_{i,j-1}\) is given by \(B = P[\alpha]TP^{T}[\beta]\). The beautiful factorization of matrix \(C\) whose entries satisfy \(c_{i,j} = \alpha c_{i-1,j} + \beta c_{i-1,j-1} + Ye_{i-1,j-1}\) is founded to be \(C = P[\alpha]DP^T[\beta]\), where \(T\) is a Toeplitz matrix, and \(P\) and \(P[\alpha]\) are Pascal matrices. The matrix product factorization to the problem is solved perfectly so far.

Dirac showed that a \(2\)-connected graph of order \(n\) with minimum degree \(\delta\) has circumference at least \(\min\{2\delta, n\}\). We prove that a \(2\)-connected, triangle-free graph \(G\) of order \(n\) with minimum degree \(\delta\) either has circumference at least \(\min\{4\delta – 4, n\}\), or every longest cycle in \(G\) is dominating. This result is best possible in the sense that there exist bipartite graphs with minimum degree \(\delta\) whose longest cycles have length \(4\delta – 4\), and are not dominating.

The vertex linear arboricity \(vla(G)\) of a graph \(G\) is the minimum number of subsets into which the vertex set \(V(G)\) can be partitioned so that each subset induces a subgraph whose connected components are paths. It is proved here that \(\lceil \frac{\omega(G)}{2}\rceil \leq vla(G) \leq \lceil \frac{\omega(G)+1}{2}\rceil\) for a claw-free connected graph \(G\) having \(\Delta(G) \leq 6\), where \(\omega(G)\) is the clique number of \(G\).

An \(f\)-coloring of a graph \(G\) is a coloring of edges of \(E(G)\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\) and denoted by \(\chi’_f(G)\). Any simple graph \(G\) has the \(f\)-chromatic index equal to \(\Delta_f(G)\) or \(\Delta_f(G) + 1\), where \(\Delta_f(G) = \max_{v \in V}\{\lceil \frac{d(v)} {f(v)}\rceil\}\). If \(\chi’_f(G) = \Delta_f(G)\), then \(G\) is of \(C_f\) \(1\); otherwise \(G\) is of \(C_f\) \(2\). In this paper, two sufficient conditions for a regular graph to be of \(C_f\) \(1\) or \(C_f\) \(2\) are obtained and two necessary and sufficient conditions for a regular graph to be of \(C_f\) \(1\) are also presented.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\), and let \(A\) be an abelian group. A labeling \(f: V(G) \to A\) induces an edge labeling \(f^*: E(G) \to A\) defined by \(f^*(xy) = f(x) + f(y)\), for each edge \(xy \in E(G)\). For \(i \in A\), let \(v_f(i) = \text{card}\{v \in V(G): f(v) = i\}\) and \(e_f(i) = \text{card}\{e \in E(G): f^*(e) = i\}\). Let \(c(f) = \{|e_f(i) – e_f(j)|: (i,j) \in A \times A\}\). A labeling \(f\) of a graph \(G\) is said to be \(A-friendly\) if \(|v_f(i) – v_f(j)| \leq 1\) for all \((i,j) \in A \times A\). If \(c(f)\) is a \((0,1)\)-matrix for an \(A\)-friendly labeling \(f\), then \(f\) is said to be \(A\)-cordial. When \(A = \mathbb{Z}_2\), the \({friendly index set}\) of the graph \(G\), \(FI(G)\), is defined as \(\{|e_f(0) – e_f(1)|: \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\}\). In this paper, we determine the friendly index set of cycles, complete graphs, and some bipartite graphs.

In this paper, several constructions are presented for balanced incomplete block designs with nested rows and columns. Some of them refine theorems due to Hishida and Jimbo \([6]\) and Uddin and Morgan \([17]\), and some of them give parameters which have not been available before.

A vertex-distinguishing edge-coloring (VDEC) of a simple graph \(G\) which contains no more than one isolated vertex and no isolated edge is equitable (VDEEC) if the absolute value of the difference between the number of edges colored by color \(i\) and the number of edges colored by color \(j\) is at most one. The minimal number of colors needed such that \(G\) has a VDEEC is called the vertex-distinguishing equitable chromatic index of \(G\). In this paper, we propose two conjectures after investigating VDEECs on some special families of graphs, such as the stars, fans, wheels, complete graphs, complete bipartite graphs, etc.

The eccentricity \(e(v)\) of a vertex \(v\) in a strongly connected digraph \(G\) is the maximum distance from \(v\). The eccentricity sequence of a digraph is the list of eccentricities of its vertices given in non-decreasing order. A sequence of positive integers is a digraphical eccentric sequence if it is the eccentricity sequence of some digraph. A set of positive integers \(S\) is a digraphical eccentric set if there is a digraph \(G\) such that \(S = \{e(v), v \in V(G)\}\). In this paper, we present some necessary and sufficient conditions for a sequence \(S\) to be a digraphical eccentric sequence. In some particular cases, where either the minimum or the maximum value of \(S\) is fixed, a characterization is derived. We also characterize digraphical eccentric sets.

Let \(C_m\) be a cycle on \(m (\geq 3)\) vertices and let \(\ominus_{n-m}C_m\) denote the class of graphs obtained from \(C_m\) by adding \(n-m (\geq 1)\) distinct pendent edges to the vertices of \(C_m\). In this paper, it is proved that for every \(T\) in \(\ominus_{n-m}C_m\), the complete graph \(K_{2n+1}\) can be cyclically decomposed into the isomorphic copies of \(T\). Moreover, if \(m\) is even, then for every positive integer \(p\), the complete graph \(K_{2pn+1}\) can also be cyclically decomposed into the isomorphic copies of \(T\).

An aperiodic perfect map (APM) is an array with the property that each possible array of a given size, called a window, arises exactly once as a contiguous subarray in the array. In this paper, we give a construction method of an APM being a proper concatenation of some fragments of a given de Bruijn sequence. Firstly, we give a criterion to determine whether a designed sequence \(T\) with entries from the index set of a de Bruijn sequence can generate an APM. This implies a sufficient condition for being an APM. Secondly, two infinite families of APMs are given by constructions of corresponding sequences \(T\), respectively, satisfying the criterion.

We introduce a combinatorial shifting operation on multicomplexes that carries similar properties required for the ordinary shifting operation on simplicial complexes. A linearly colored simplicial complex is called shifted if its associated multicomplex is stable under defined operation. We show that the underlying simplicial subcomplex of a linearly shifted simplicial complex is shifted in the ordinary sense, while the ordinary and linear shiftings are not interrelated in general. Separately, we also prove that any linearly shifted complex must be shellable with respect to the order of its facets induced by the linear coloring. As an application, we provide a characterization of simple graphs whose independence complexes are linearly shifted. The class of graphs obtained constitutes a superclass of threshold graphs.

A local coloring of a graph \(G\) is a function \(c: V(G) \to \mathbb{N}\) having the property that for each set \(S \subseteq V(G)\) with \(2 \leq |S| \leq 3\), there exist vertices \(u,v \in S\) such that \(|c(u) – c(v)| \geq m_S\), where \(m_S\) is the size of the induced subgraph \(\langle S\rangle\). The maximum color assigned by a local coloring \(c\) to a vertex of \(G\) is called the value of \(c\) and is denoted by \(\chi_\ell(c)\). The local chromatic number of \(G\) is \(\chi_\ell(G) = \min\{\chi_\ell(c)\}\), where the minimum is taken over all local colorings \(c\) of \(G\). If \(\chi_\ell(c) = \chi_\ell(G)\), then \(c\) is called a minimum local coloring of \(G\). The local coloring of graphs introduced by Chartrand et al. in \(2003\). In this paper, following the study of this concept, first an upper bound for \(\chi_\ell(G)\) where \(G\) is not complete graphs \(K_4\) and \(K_5\), is provided in terms of maximum degree \(\Delta(G)\). Then the exact value of \(\chi_\ell(G)\) for some special graphs \(G\) such as the cartesian product of cycles, paths and complete graphs is determined.

Explicit expressions for all the primitive idempotents in the ring \(R_{2^n} = {F}_q[x]/(x^{2^n} – 1)\), where \(q\) is an odd prime power, are obtained. Some lower bounds on the minimum distances of the irreducible cyclic codes of length \(2^n\) over \({F}_q\) are also obtained.

In this note we prove that all connected Cayley graphs of every finite group \(Q \times H\) are \(1\)-factorizable, where \(Q\) is any non-trivial group of \(2\)-power order and \(H\) is any group of odd order.

A graph \(G\) is called super vertex-magic total labelings if there exists a bijection \(f\) from \(V(G) \cup E(G)\) to \(\{1,2,\ldots,|V(G)| + |E(G)|\}\) such that \(f(v) + \sum_{u \sim v} f(vu) = C\), where the sum is over all vertices \(u\) adjacent to \(v\) and \(f(V(G)) = \{1,2,\ldots,|V(G)|\}\), \(f(E(G)) = \{|V(G)|+1,|V(G)|+2,\ldots,|V(G)|+|E(G)|\}\). \({The Knödel graphs}\) \(W_{\Delta,n}\) have even \(n \geq 2\) vertices and degree \(\Delta\), \(1 \leq \Delta \leq \lfloor\log_2 n\rfloor\). The vertices of \(W_{\Delta,n}\) are the pairs \((i,j)\) with \(i = 1,2\) and \(0 \leq i \leq n/2-1\). For every \(j\), \(0 \leq j \leq n/2-1\), there is an edge between vertex \((1,j)\) and every vertex \((2,(j+2^k-1) \mod (n/2))\), for \(k=0,\ldots,\Delta-1\). In this paper, we show that \(W_{3,n}\) is super vertex-magic for \(n \equiv 0 \mod 4\).

Evolutionary graphs were initially proposed by Lieberman \(et \;al\). and evolutionary dynamics on two levels are recently introduced by Traulsen et al. We now introduce a new type of evolutionary dynamics,evolutionary graphs on two levels, and the fixation probability is analyzed. Some interesting results, evolutionary graphs on two levels are more stable than single level evolutionary graphs, are obtained in this paper.

A vertex \(k\)-ranking of a graph \(G\) is a function \(c: V(G) \to \{1,\ldots,k\}\) such that if \(c(u) = c(v)\), \(u,v \in V(G)\), then each path connecting vertices \(u\) and \(v\) contains a vertex \(w\) with \(c(w) > c(u)\). If each vertex \(v\) has a list of integers \(L(v)\) and for a vertex ranking \(c\) it holds \(c(v) \in L(v)\) for each \(v \in V(G)\), then \(c\) is called an \(L\)-list \(k\)-ranking, where \(\mathcal{L} = \{L(v) : v \in V(G)\}\). In this paper, we investigate both vertex and edge (vertex ranking of a line graph) list ranking problems. We prove that both problems are NP-complete for several classes of acyclic graphs, like full binary trees, trees with diameter at most \(4\), and comets. The problem of finding vertex (edge) \(\mathcal{L}\)-list ranking is polynomially solvable for paths and trees with a bounded number of non-leaves, which includes trees with diameter less than \(4\).

In this paper we determine unique graph with largest spectral radius among all tricyclic graphs with \(n\) vertices and \(k\) pendant edges.

A new proof is given to the following result of ours. Let \(G\) be an outerplanar graph with maximum degree \(\Delta \geq 3\). The chromatic number \(\chi(G^2)\) of the square of \(G\) is at most \(\Delta+2\), and \(\chi(G^2) = \Delta+1\) if \(\Delta \geq 7\).

Some designs using the action of the linear fractional groups \(L_2(q)\), \(q = 11, 13, 16, 17, 19, 23\) are constructed. We will show that \(L_2(q)\) or its automorphism group acts as the full automorphism group of each of the constructed designs except in the case \(q = 16\). For designs constructed from \(L_2(16)\), we will show that \(L_2(16)\), \(L_2(16) : 2\), \(L_2(16) : 4\) or \(S_{17}\) can arise as the full automorphism group of the design.

For odd \(n \geq 5\), the Flower Snark \(F_n = (V, E)\) is a simple undirected cubic graph with \(4n\) vertices, where \(V = \{a_i : 0 \leq i \leq n-1\} \cup \{b_i : 0 \leq i \leq n-1\} \cup \{c_i : 0 \leq i \leq 2n-1\}\) and \(E = \{b_ib_{(i+1)\mod(n)}: 0 \leq i \leq n-1\} \cup \{c_ic_{(i+1)\mod(2n)} : 0 \leq i \leq 2n-1\} \cup \{a_ib_i,a_ic_i,a_ic_{n+i} : 0 \leq i \leq n-1\}\). For \(n = 3\) or even \(n \geq 4\), \(F_n\) is called the related graph of Flower Snark. We show that the crossing number of \(F_n\) equals \(n – 2\) if \(3 \leq n \leq 5\), and \(n\) if \(n \geq 6\).

A subset \(S\) of the vertex set of a graph \(G\) is called acyclic if the subgraph it induces in \(G\) contains no cycles. We call \(S\) an acyclic dominating set if it is both acyclic and dominating. The minimum cardinality of an acyclic dominating set, denoted by \(\gamma_a(G)\), is called the acyclic domination number of \(G\). A graph \(G\) is \({2-diameter-critical}\) if it has diameter \(2\) and the deletion of any edge increases its diameter. In this paper, we show that for any positive integers \(k\) and \(d \geq 3\), there is a \(2\)-diameter-critical graph \(G\) such that \(\delta(G) = d\) and \(\gamma_a(G) – \delta(G) \geq k\), and our result answers a question posed by Cheng et al. in negative.

A function \(f: V \to \{1,\ldots,k\}\) is a broadcast coloring of order \(k\) if \(\pi(u) = \pi(v)\) implies that the distance between \(u\) and \(v\) is more than \(\pi(u)\). The minimum order of a broadcast coloring is called the broadcast chromatic number of \(G\), and is denoted \(\chi_b(G)\). In this paper we introduce this coloring and study its properties. In particular, we explore the relationship with the vertex cover and chromatic numbers. While there is a polynomial-time algorithm to determine whether \(\chi_b(G) \leq 3\), we show that it is \(NP\)-hard to determine if \(\chi_b(G) \leq 4\). We also determine the maximum broadcast chromatic number of a tree, and show that the broadcast chromatic number of the infinite grid is finite.

A connected graph \(G = (V, E)\) is said to be \((a,d)\)-antimagic if there exist positive integers \(a,d\) and a bijection \(f : E \to \{1,2,\ldots,|E|\}\) such that the induced mapping \(g_f : V \to \mathbb{N}\), defined by \(g_f(v) = \sum f(uv)\),\({uv \in E(G)}\) is injective and \(g_f(V) = \{a,a+d,\ldots,a+(|V|-1)d\}\). Mirka Miller and Martin Bača proved that the generalized Petersen graph \(P(n, 2)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\) and conjectured that the generalized Petersen graph \(P(n, k)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). In this paper, we show that the generalized Petersen graph \(P(n, 3)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for even \(n \geq 8\).

In this paper, we derive new recurrence relations and generating matrices for the sums of usual Tribonacci numbers and \(4n\) subscripted Tribonacci sequences, \(\{T_{4n}\}\), and their sums. We obtain explicit formulas and combinatorial representations for the sums of terms of these sequences. Finally, we represent relationships between these sequences and permanents of certain matrices.

Let \(\mathcal{K} = (K_{ij})\) be an infinite lower triangular matrix of non-negative integers such that \(K_{i0} = 1\) and \(K_{ii} \geq 1\) for \(i \geq 0\). Define a sequence \(\{V_i(\mathcal{K})\}_{m\geq0}\) by the recurrence \(V_{i+1}(\mathcal{K}) = \sum_{j=0}^m K_{mj}V_j(\mathcal{K})\) with \(V_0(\mathcal{K}) = 1\). Let \(P(n;\mathcal{K})\) be the number of partitions of \(n\) of the form \(n = p_1 + p_2 + p_3 + p_4 + \cdots\) such that \(p_j \geq \sum_{i\geq j} K_{ij}p_{i+1}\) for \(j \geq 1\) and let \(P(n;V(\mathcal{K}))\) denote the number of partitions of \(n\) into summands in the set \(V(\mathcal{K}) = \{V_1(\mathcal{K}), V_2(\mathcal{K}), \ldots\}\). Based on the technique of MacMahon’s partitions analysis, we prove that \(P(n;\mathcal{K}) = P(n;V(\mathcal{K}))\) which generalizes a recent result of Sellers’. We also give several applications of this result to many classical sequences such as Bell numbers, Fibonacci numbers, Lucas numbers, and Pell numbers.