In this note, we consider one type of \(k\)-tridiagonal matrix family whose permanents and determinants are specified to the balancing and Lucas-balancing numbers. Moreover, we provide some properties between Chebyshev polynomial properties and the given number
sequences,

Let \(G = (V, E)\) be a graph. A subset \(D \subseteq V\) is a dominating set if every vertex not in \(D\) is adjacent to a vertex in \(D\). The domination number of \(G\) is the smallest cardinality of a dominating set of \(G\). The bondage number of a nonempty graph \(G\) is the smallest number of edges whose removal from \(G\) results in a graph with larger domination number than \(G\). In this paper, we determine that the exact value of the bondage number of an \((n-3)\)-regular graph \(G\) of order \(n\) is \(n-3\).

A graph is closed when its vertices have a labeling by \([n]\) with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.

Let \(\Sigma = (X, \mathcal{B})\) be a \(4\)-cycle system of order \(v = 1 + 8k\). A \(c\)-colouring of type \(s\) is a map \(\phi: \mathcal{B} \to C\), where \(C\) is a set of colours, such that exactly \(c\) colours are used and for every vertex \(x\), all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(4k = qs + r\), with \(r \geq 0\). \(\phi\) is equitable if for every vertex \(x\), the set of the \(4k\) blocks containing \(x\) is partitioned into \(r\) colour classes of cardinality \(q + 1\) and \(s – r\) colour classes of cardinality \(q\). In this paper, we study colourings for which \(s | k\), providing a description of equitable block colourings for \(c \in \{s, s+1, \ldots, \lfloor \frac{2s^2+s}{3} \rfloor\}\).

In this paper, we first introduce a linear program on graphical invariants of a graph \(G\). As an application, we attain the extremal graphs with lower bounds on the first Zagreb index \(M_1(G)\), the second Zagreb index \(M_2(G)\), their multiplicative versions \(\Pi_1^*(G)\), \(\Pi_2(G)\), and the atom-bond connectivity index \(ABC(G)\), respectively.

Let \(\Gamma\) be an oriented graph. We denote the in-neighborhood and out-neighborhood of a vertex \(v\) in \(\Gamma\) by \(\Gamma^-(v)\) and \(\Gamma^+(v)\), respectively. We say \(\Gamma\) has Property \(A\) if, for each arc \((u,v)\) in \(\Gamma\), each of the graphs induced by \(\Gamma^+(u) \cap \Gamma^+(v)\), \(\Gamma^-(u) \cap \Gamma^-(v)\), \(\Gamma^-(u) \cap \Gamma^+(v)\), and \(\Gamma^+(u) \cap \Gamma^-(v)\) contains a directed cycle. Moreover, \(\Gamma\) has Property B if each arc \((u,v)\) in \(\Gamma\) extends to a \(3\)-path \((x,u), (u,v), (v,w)\), such that \(|\Gamma^+(x) \cap \Gamma^+(u)| \geq 5\) and \(|\Gamma^-(v) \cap \Gamma^-(w)| \geq 5\). We show that the only oriented graphs of order at most \(17\), which have both properties \(A\) and \(B\), are the Tromp graph \(T_{16}\) and the graph \(T^+_{16}\), obtained by duplicating a vertex of \(T_{16}\). We apply this result to prove the existence of an oriented planar graph with oriented chromatic number at least \(18\).

By the partial fraction decomposition method, we establish a \(q\)-harmonic sum identity with multi-binomial coefficient, from which we can derive a fair number of harmonic number identities.

A fall \(k\)-coloring of a graph \(G\) is a proper \(k\)-coloring of \(G\) such that each vertex of \(G\) sees all \(k\) colors on its closed neighborhood. We denote \(\text{Fall}(G)\) the set of all positive integers \(k\) for which \(G\) has a fall \(k\)-coloring. In this paper, we study fall colorings of the lexicographic product of graphs and the categorical product of graphs. Additionally, we show that for each graph \(G\), \(\text{Fall}(M(G)) = \emptyset\), where \(M(G)\) is the Mycielskian of the graph \(G\). Finally, we prove that for each bipartite graph \(G\), \(\text{Fall}(G^c) \subseteq \{\chi(G^c)\}\) and it is polynomial time to decide whether \(\text{Fall}(G^c) = \{\chi(G^c)\}\) or not.

In this paper, we first provide two necessary conditions for a graph \(G \) to be \(E_k\)-cordial, then we prove that every \(P_n(n \geq 3)\) is \(E_p\)-cordial if \(p\) is odd. In the end, we discuss the \(E_2\)-cordiality of a graph \)G\) under the condition that some subgraph of \(G\) has a \(1\)-factor.