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.

In this paper, we consider the problem of determining the structure of a minimal critical set in a latin square \(L\) representing the elementary abelian \(2\)-group of order \(8\).

In this paper, the first two (resp. four) largest signless Laplacian spectral radii together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order n are determined, and the first two (resp. four) largest signless Laplacian spreads together with the corresponding graphs in the class of bicyclic (resp. tricyclic) graphs of order \(n\) are identified.