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.

An edge-magic total labeling of a graph \(G\) is a one-to-one map \(\lambda\) from \(V(G) \cup E(G)\) onto the integers \(\{1, 2, \ldots, |V(G) \cup E(G)|\}\) with the property that there exists an integer constant \(c\) such that \(\lambda(x) + \lambda(x,y) + \lambda(y) = c\) for any \((x, y) \in E(G)\). If \(\lambda(V(G)) = \{1, 2, \ldots, |V(G)|\}\), then the edge-magic total labeling is called super edge-magic total labeling. In this paper, we formulate super edge-magic total labeling on subdivisions of stars \(K_{1,p}\), for \(p \geq 5\).

In this paper, we briefly survey Euler’s works on identities connected with his famous Pentagonal Number Theorem. We state a partial generalization of his theorem for partitions with no part exceeding an identified value \(k\), along with some identities linking total partitions to partitions with distinct parts under the above constraint. We derive both recurrence formulas and explicit forms for \(\Delta_n(m)\), where \(\Delta_n(m)\) denotes the number of partitions of \(m\) into an even number of distinct parts not exceeding \(n\), minus the number of partitions of \(m\) into an odd number of distinct parts not exceeding \(n\). In fact, Euler’s Pentagonal Number Theorem asserts that for \(m \leq n\), \(\Delta_n(m) = \pm 1\) if \(m\) is a Pentagonal Number and \(0\) otherwise. Finally, we establish two identities concerning the sum of bounded partitions and their connection to prime factors of the bound integer.