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.