For a connected graph \(G\) and any two vertices \(u\) and \(v\) in \(G\), let \(d(u,v)\) denote the distance between \(u\) and \(v\) and let \(d(G)\) be the diameter of \(G\). For a subset \(S\) of \(V(G)\), the distance between \(v\) and \(S\) is \(d(v, S) = \min\{d(v,x) \mid x \in S\}\). Let \(\Pi = \{S_1, S_2, \ldots, S_k\}\) be an ordered \(k\)-partition of \(V(G)\). The representation of \(v\) with respect to \(\Pi\) is the \(k\)-vector \(r(v \mid \Pi) = (d(v, S_1), d(v, S_2), \ldots, d(v, S_k))\). A partition \(\Pi\) is a resolving partition for \(G\) if the \(k\)-vectors \(r(v \mid \Pi)\), \(v \in V(G)\) are distinct. The minimum \(k\) for which there is a resolving \(k\)-partition of \(V(G)\) is the partition dimension of \(G\), and is denoted by \(pd(G)\). A partition \(\Pi = \{S_1, S_2, \ldots, S_k\}\) is a resolving path \(k\)-partition for \(G\) if it is a resolving partition and each subgraph induced by \(S_i\), \(1 \leq i \leq k\), is a path. The minimum \(k\) for which there exists a path resolving \(k\)-partition of \(V(G)\) is the path partition dimension of \(G\), denoted by \(ppd(G)\). In this paper, path partition dimensions of trees and the existence of graphs with given path partition, partition, and metric dimension, respectively, are studied.

Let \(A\) be an abelian group with \(|A| \geq 4\). Suppose that \(G\) is a \(3\)-edge-connected simple graph on \(n \geq 19\) vertices. We show in this paper that if \(\max\{d(x), d(y), d(z)\} \geq n/6\) for every \(3\)-independent vertices \(\{x, y, z\}\) of \(G\), then either \(G\) is \(A\)-connected or \(G\) can be \(T\)-reduced to the Petersen graph, which generalizes the result of Zhang and Li (Graphs and Combin., \(30 (2014), 1055-1063).\)

Let \({F}_q\) be a finite field of odd order \(q\). In this note, the generator polynomials and the numbers of all self-dual and self-orthogonal cyclic and negacyclic codes of length \(2^m\) over \({F}_q\) are precisely characterized.

In this paper, we find the star chromatic number \(\chi_s\) for the central graph of sunlet graphs \(C(S_n)\), line graph of sunlet graphs \(L(S_n)\), middle graph of sunlet graphs \(M(S_n)\), and the total graph of sunlet graphs \(T(S_n)\).

Multireceiver authentication codes allow one sender to construct an authenticated message for a group of receivers such that each receiver can verify the authenticity of the received message. In this paper, we construct multireceiver authentication codes from pseudo-symplectic geometry over finite fields. The parameters and the probabilities of deceptions of the two codes are also computed.

For a simple undirected graph \(G\) with vertex set \(V\) and edge set \(E\), a total \(k\)-labeling \(\lambda: V \cup E \rightarrow \{1, 2, \ldots, k\}\) is called a vertex irregular total \(k\)-labeling of \(G\) if for every two distinct vertices \(x\) and \(y\) of \(G\), their weights \(wt(x)\) and \(wt(y)\) are distinct, where the weight of a vertex \(x\) in \(G\) is the sum of the label of \(x\) and the labels of all edges incident with the vertex \(x\). The total vertex irregularity strength of \(G\), denoted by \(\text{tus}(G)\), is the minimum \(k\) for which the graph \(G\) has a vertex irregular total \(k\)-labeling. The complete \(m\)-partite graph on \(n\) vertices in which each part has either \(\left\lfloor \frac{n}{m} \right\rfloor\) or \(\left\lceil \frac{n}{m} \right\rceil\) vertices is denoted by \(T_{n,m}\). The total vertex irregularity strength of some equitable complete \(m\)-partite graphs, namely, \(T_{m,m+1}\), \(T_{m,m+2}\), \(T_{m,2m}\), \(T_{m,2m+4}\), \(T_{3m-1}\) (\(m \geq 4\)), \(T_{n}\) (\(n = 3m+r\), \(r = 1, 2, \ldots, m-1\)), and equitable complete \(3\)-partite graphs have been studied in this paper.

Suppose \(m\) and \(t\) are integers such that \(0 < t \leq m\). An \((m,t)\)-splitting system is a pair \((X, \mathcal{B})\) that satisfies for every \(Y \subseteq X\) with \(|Y| = t\), there is a subset \(B\) of \(X\) in \(\mathcal{B}\), such that \(|B \cap Y| = \left\lfloor \frac{t}{2} \right\rfloor\) or \(|(X \setminus B) \cap Y| = \left\lceil \frac{t}{2} \right\rceil\). Suppose \(m\), \(t_1\), and \(t_2\) are integers such that \(t_1 + t_2 \leq m\). An \((m, t_1, t_2)\)-separating system is a pair \((X, \mathcal{B})\) which satisfies for every \(P \subseteq X\), \(Q \subseteq X\) with \(|P| = t_1\), \(|Q| = t_2\), and \(P \cap Q = \emptyset\), there exists a block \(B \in \mathcal{B}\) for which either \(P \subseteq B\), \(Q \cap B = \emptyset\) or \(Q \subseteq B\), \(P \cap B = \emptyset\). We will give some results on splitting systems and separating systems for \(t = 5\) and \(t = 6\).

Motivated by the recent work by Ramirez \([8]\), related to the bi-periodic Fibonacci sequences, here we introduce the bi-periodic incomplete Lucas sequences that gives the incomplete Lucas sequence as a special case. We also give recurrence relations and the generating function of these sequences. Also, we give a relation between bi-periodic incomplete Fibonacci sequences and bi-periodic incomplete Lucas sequences.

In this paper, we prove the \(q\)-log-convexity of Domb’s polynomials, which was conjectured by Sun in the study of series for powers of \(\pi\). As a result, we obtain the log-convexity of Domb’s numbers. Our proof is based on the \(q\)-log-convexity of Narayana polynomials of type \(B\) and a criterion for determining \(q\)-log-convexity of self-reciprocal polynomials.

Two Schwenk-like formulas about the signless Laplacian matrix of a graph are given, and thus it gives new tools for computing \(Q\)-
characteristic polynomials of graphs directly. As an application, we give the \(Q\)-characteristic polynomial of lollipop graphs and reprove the known result that no two non-isomorphic lollipop graphs are \(Q\)-cospectral by a simple manner.