Let \(G\) be a simple graph of order \(n\). We define a dominating set as a set \(S \subseteq V(G)\) such that every vertex of \(G\) is either in \(S\) or adjacent to a vertex in \(S\). The \({domination\; polynomial}\) of \(G\) is \(D(G, x) = \sum_{i=0}^{n} d(G, i)x^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\). Two graphs \(G\) and \(H\) are \({D-equivalent}\), denoted \(G \sim H\), if \(D(G, x) = D(H, x)\). The \({D-equivalence\; class}\) of \(G\) is \([G] = \{H \mid H \sim G\}\). Recently, determining the \(D\)-equivalence class of a given graph has garnered interest. In this paper, we show that for \(n \geq 3\), \([K_{n,n}]\) has size two. We conjecture that the complete bipartite graph \(K_{m,n}\) for \(m, n \geq 2\) is uniquely determined by its domination polynomial.

The Jacobi matrix polynomials and their orthogonality only for commutative matrices was first studied by Defez \(et. al\).
[Jacobi matrix differential equation, polynomial solutions and their properties. Comput. Math. Appl. \(48 (2004), 789-803]\). It is known that orthogonal matrix polynomials comprise an emerging field of study, with important results in both theory and applications continuing to appear in the literature. The main object of this paper is to derive various families of linear, multilateral and multilinear generating functions for the Jacobi matrix polynomials and the Gegenbauer matrix polynomials. Recurrence relations of Jacobi matrix polynomials are obtained. Some special cases of the results presented in this study are also indicated.

The positive index of inertia of a signed graph \(\Gamma\), denoted by \(,(\Gamma)\), is the number of positive eigenvalues of the adjacency matrix \(A(\Gamma)\) including multiplicities. In this paper we investigate the minimal positive index of inertia of
signed unicyclic graphs of order \(n\) with fixed girth and characterize the extremal graphs with the minimal positive index. Finally, we characterize the signed unicyclic graphs with the positive indices \(1\) and \(2\).

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.