Let \( G \) be a graph with vertex set \( V \) and no isolated vertices. A subset \( S \subseteq V \) is a semipaired dominating set of \( G \) if every vertex in \( V \setminus S \) is adjacent to a vertex in \( S \) and \( S \) can be partitioned into two-element subsets such that the vertices in each subset are at most distance two apart. We present a method of building trees having a unique minimum semipaired dominating set.

We discuss the connections tying Laplacian matrices to abstract duality and planarity of graphs.

A set \( S \subseteq V(G) \) of a connected graph \( G \) is a co-secure dominating set, if \( S \) is a dominating set and for each \( u \in S \), there exists a vertex \( v \in V(G) \setminus S \), such that \( v \in N(u) \) and \( (S \setminus \{u\}) \cup \{v\} \) is a dominating set of \( G \). The minimum cardinality of the co-secure dominating set in a graph \( G \) is the co-secure domination number, \( \gamma_{cs}(G) \). In this paper, we characterise the Mycielski graphs with co-secure domination 2 and 3. We also obtained a sharp upper bound for \( \gamma_{cs}(G) \).

Given an n-connected binary matroid, we obtain a necessary and sufficient condition for its single-element coextensions to be \(n\)-connected.

In this paper, we provide bounds for the crossing number of mesh connected trees and 3-regular mesh of trees.

The ordinary factorial may be written in terms of the Stirling numbers of the second kind as shown by Quaintance and Gould and the odd double factorial in terms of the Stirling numbers of the first kind as shown by Callan. During the preparation of an expository paper on Wolstenholme’s Theorem, we discovered an expression for the odd double factorial using the Stirling numbers of the second kind. This appears to be the first such identity involving both positive and negative integers and the result is outlined here.

Let \( G \) be a \( k \)-connected (\( k > 2 \)) graph of order \( n \). If \( \chi(G) > n-k \), then \( G \) is Hamiltonian or \( K_k \vee (K_{n-2k}) \) with \( n > 2k+1 \), where \( \chi(G) \) is the chromatic number of the graph \( G \).

An \((n, r)\)-arc in \(PG(2, q)\) is a set of \(n\) points such that each line contains at most \(r\) of the selected points. We show that in the case of the existence of a \((101, 10)\)-arc in \(PG(2, 11)\) it only admits the trivial linear automorphism.

In this paper, we generalise the notion of distance irregular labeling introduced by Slamin to vertex irregular \(d\)-distance vertex labeling, for any distance \(d\) up to the diameter. We also define the inclusive vertex irregular \(d\)-distance vertex labeling. We give the lower bound of the inclusive vertex irregular \(1\)-distance vertex labeling for general graphs and a better lower bound on caterpillars. The inclusive labelings for paths \(P_n, n \equiv 0 \mod 3\), stars \(S_n\), double stars \(S(m,n)\), cycles \(C_n\), and wheels \(W_n\) are provided. From the inclusive vertex irregular \(1\)-distance vertex labeling on cycles, we derive the vertex irregular \(1\)-distance vertex labeling on prisms.

A Hamiltonian walk in a nontrivial connected graph \( G \) is a closed walk of minimum length that contains every vertex of \( G \). The 3-path graph \( P_3(G) \) of a connected graph \( G \) of order 3 or more has the set of all 3-paths (paths of order 3) of \( G \) as its vertex set and two vertices of \( P_3(G) \) are adjacent if they have a 2-path in common. With the aid of Hamiltonian walks in spanning trees of the 3-path graph of a graph, it is shown that if \( G \) is a connected graph with minimum degree at least 4, then \( P_3(G) \) is Hamiltonian-connected.