In this paper, we explicitly explore the endomorphism monoid of the circulant complete graph \(K(n, 4)\). We demonstrate that \(Aut(K(n,4)) \cong D_n\), the dihedral group of degree \(n\). Furthermore, we show that \(K(n,4)\cong D_n\) is unretractive for \(n = 4m , 4m +2\) (\(m \geq 2\)), and that \(End(K(n,4)) = qEnd(K(n,4))\) and \(sEnd(K(n,4)) = Aut(K(n,4))\) when \(n = 4m, 4m + 2\) (\(m \geq 2\)). Additionally, we prove that \(End(K(4m,4))\) is regular and \(End(K(4m + 2,4))\) is completely regular. We also solve some enumerative problems concerning \(End(K(n,4))\) are solved.

In this note we find a necessary and sufficient condition for the supersolvability of an essential, central arrangement of rank \(3\) (\(i.e\), line arrangement in the projective plane). We present an algorithmic way to decide if such an arrangement is supersoivable or not that does not require an ordering of the lines as the Bjémer-Ziegler’s and Peeva’s criteria require. The method uses the duality between points and lines in the projective plane in the context of coding theory.

This paper deals with the Abelian sandpile model on the generalized trees with certain given boundary condition. Using a combinatorial method, we obtain the exact expressions for all single-site probabilities and some two-site joint probabilities. Also, we prove that the sites near the boundary have a different height probability from those away from it in bulk for the Bethe lattice with the boundary condition, which is the same as those results found by Grassberger and Manna [Some more sandpiles,” J.Phys.(France)\(51,1077-1098(1990)\)] and proved by Haiyan chen and Fuji Zhang [“Height probabilities in the Abelian sandpile on the generalized finite Bethe lattice” J. Math. Phys. \(54, 083503 (2013))\).

Let \(S\) be a subset of the positive integers and \(M\) be a positive integer. Inspired by Tony Colledge’s work, Mohammad K. Azarian considered the number of ways to climb a staircase with \(n\) stairs using “step-sizes” \(s \in S\) with multiplicities at most \(M\). In this exposition, we find a solution via generating functions, i.e., an expression counting the number of partitions \(n = \sum_{s \in S} m_ss\), satisfying \(0 \leq m_s \leq M\). We then use this result to answer a series of questions posed by Azarian, establishing a link with ten sequences listed in the On-Line Encyclopedia of Integer Sequences (OEIS). We conclude by posing open questions that seek to count the number of compositions of \(n\).

Hamiltonian index of a graph \(G\) is the smallest positive integer \(k\), for which the \(k\)-th iterated line graph \(L^k(G)\) is hamiltonian. Bedrossian characterized all pairs of forbidden induced subgraphs that imply hamiltonicity in \(2\)-connected graphs. In this paper, some upper bounds on the hamiltonian index of a \(2\)-connected graph in terms of forbidden not necessarily induced subgraphs are presented.

The Szeged polynomial of a connected graph \(G\) is defined as \(S_z(G,x) = \sum_{e \in E(G)} x^{n_{u(e) n_v(e)}} \), where \(n_u(e)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\), and \(n_v(e)\) is the number of vertices of \(G\) lying closer to \(v\) than to \(u\). Ashrafi et al. (On Szeged polynomial of a graph, Bull. Iran. Math. Soc. \(33 (2007) 37-46)\) proved that if \(|V(G)|\) is even, then \(\deg(S_z(G,x)) \leq \frac{1}{4}{|V(G)^{2}} |\). In this paper, we investigate the structure of graphs with an even number of vertices for which equality holds, and also examine equality for the sum of graphs.

In this paper we investigate the number of rooted loopless unicursal planar maps and present some formulae for such maps with up to three parameters: the number of edges and the valencies of the two odd vertices.

In this paper, we investigate the metric dimension of generalized Petersen graphs \(P(n,3)\), providing a partial answer to an open problem posed in [8]: whether \(P(n,m)\) for \(n \geq 7\) and \(3 \leq m \leq \left\lfloor \frac{n-1}{2} \right\rfloor\) constitutes a family of graphs with constant metric dimension. Specifically, we prove that the metric dimension of \(P(n,3)\) equals \(3\) for \(n \equiv 1 \pmod{6}\), \(n \geq 25\), and equals \(4\) for \(n \equiv 0 \pmod{6}\), \(n \geq 24\). For remaining cases, four judiciously chosen vertices suffice to resolve all vertices of \(P(n,3)\), implying \(\dim(P(n,3)) \leq 4\), except when \(n \equiv 2 \pmod{6}\), in which case \(\dim(P(n,3)) \leq 5\).

Using subspaces of the finite field \(GF(q^{2^k})\) over \(GF(q)\), we construct new classes of external difference families.

Let \(M = \{v_1, v_2, \ldots, v_n\}\) be an ordered set of vertices in a graph \(G\). Then, \((d(u, v_1), d(u, v_2), \ldots, d(u, v_n))\) is called the \(M\)-coordinates of a vertex \(u\) of \(G\). The set \(M\) is called a \({metric\; basis}\) if the vertices of \(G\) have distinct \(M\)-coordinates. A minimum metric basis is a set \(M\) with minimum cardinality. The cardinality of a minimum metric basis of \(G\) is called the minimum metric dimension. This concept has wide applications in motion planning and robotics. In this paper, we solve the minimum metric dimension problem for Illiac networks.