The aim of this paper is to answer a question proposed by Li \([2]\) and prove that no connected bi-normal Cayley graph other than cycles of even length is \(3\)-arc-transitive.

Using new ways to label edges in an ordered tree, this paper introduces two bijections between bicoloured ordered trees and non-crossing partitions. Consequently, enumeration results of non-crossing partitions specified with several parameters are derived.

The first and second multiplicative Zagreb indices of a simple graph \(G\) are defined as:
\[ \prod_1(G) = \prod_{u \in V(G)} d_G(u)^2
\text{and}
\prod_2(G) = \prod_{uv \in E(G)} d_G(u)d_G(v),\]
where \(d_G(u)\) denotes the degree of the vertex \(u\) of \(G\). In this paper, we establish strict lower bounds on the first and second multiplicative Zagreb indices of various graph operations in terms of the first and second multiplicative Zagreb indices and multiplicative sum Zagreb index of their components.

This paper contributes to the study of automorphism groups of \(2-(v, k, 1)\) designs. Let \(\mathcal{D}\) be a \(2-(v, 31, 1)\) design and \(G \leq Aut(\mathcal{D})\) be block-transitive and point-primitive. If \(G\) is unsolvable, then \(Soc(G)\), the socle of \(G\), is not isomorphic to \(^2F_4(q)\).

The Randić index of a graph \(G\), denoted by \(R(G)\), is defined as the sum of \(\frac{1}{d(u)d(v)}\) over all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). Denote by \(\nu(G)\) the matching number, i.e., the number of edges in a maximum matching of \(G\). A conjecture of AutoGraphiX on the relation between the Randić index and the matching number of a connected graph \(G\) states: for any connected graph of order \(n \geq 3\) with Randić index \(R(G)\) and matching number \(\mu(G)\),
\[ R(G) – \mu(G) \leq \sqrt{\lfloor\frac{n+4}{7}\rfloor \lfloor \frac{6n+2}{7} \rfloor} -\lfloor \frac{n+4}{7}\rfloor \]
with equality if and only if \(G\) is a complete bipartite graph \(K_{p,q}\) with \(p = \mu(G) = \left\lfloor \frac{n+4}{2} \right\rfloor\), which was proposed by Aouchiche et al. In this paper, we confirm this conjecture for some classes of graphs.

A 2-semiarc is a pointset \(\mathcal{S}_2\) with the property that the number of tangent lines to \(\mathcal{S}_2\) at each of its points is two. Using theoretical results and computer-aided search, we provide the complete classification of 2-semiarcs in \(PG(2, q)\) for \(q \leq 7\), determine the spectrum of their sizes for \(q \leq 9\), and prove existence results for \(q = 11\) and \(q = 13\). Additionally, for several sizes of 2-semiarcs in \(PG(2, q)\) with \(q \leq 7\), classification results have been obtained through theoretical proofs.

In this paper, we concentrate on rooted general maps on all surfaces(orientable and nonorientable) without regard to genus and present the enumerating equation with respect to vertices and edges, which is a Riccati’s equation. To solve it, a new solution in continued fraction form is given. As two especial cases, the corresponding results of rooted general maps and rooted monopole maps on all surfaces with respect to edges regardless of genus are obtained.

For a tree \(T\), the set of leaves of \(T\) is denoted by \(Leaf(T)\), and the subtree \(T – Leaf(T)\) is called the \({stem} of T\). We prove that if a connected graph \(G\) either satisfies \(\sigma_{k+1}(G) \geq |G| – k – 1\) or has no vertex set of size \(k+1\) such that the distance between any two of its vertices is at least \(4\), then \(G\) has a spanning tree whose stem has at most \(k\) leaves, where \(\sigma_{k+1}(G)\) denotes the minimum degree sum of \(k+1\) independent vertices of \(G\). Moreover, we show that the condition on \(\sigma_{k+1}(G)\) is sharp. Additionally, we provide another similar sufficient degree condition for a claw-free graph to have such a spanning tree.

We prove that every connected subcubic graph G has two spanning trees \(T_1,T_2\) such that every component of \(G – E(T_1)\) is a path of length at most \(3\), and every component of \(G – E(T_2)\) is either a path of length at most \(2\) or a cycle.

A graph \(X\) is said to be \({End-completely-regular}\) (\({End-inverse}\)) if its endomorphism monoid \(End(X)\) is completely regular (inverse). In this paper, we demonstrate that if \(X + Y\) is End-completely-regular, then both \(X\) and \(Y\) are End-completely-regular. We present several approaches to construct new End-completely-regular graphs via the join of two graphs with specific conditions. Notably, we determine the End-completely-regular joins of bipartite graphs. Furthermore, we prove that \(X + Y\) is End-inverse if and only if \(X + Y\) is End-regular and both \(X\) and \(Y\) are End-inverse. Additionally, we determine the End-inverse joins of bipartite graphs.

The tensor product of two graphs \(G_1\) and \(G_2\), denoted by \(G_1 \times G_2\), is defined as the graph with vertex set \(\{(x, y): x \in V(G_1), y \in V(G_2)\}\) and edge set \(\{(x_1, y_1)(x_2, y_2): x_1x_2 \in E(G_1), y_1y_2 \in E(G_2)\}\). Very recently, Zhang, Zheng, and Mamut showed that if \(\delta(G_1) \geq 2\) and \(G_2\) does not belong to a well-characterized class \(\mathcal{G}\) of graphs, then \(G_1 \times G_2\) admits a nowhere-zero \(3\)-flow. However, it remains unclear whether \(G_1 \times G_2\) admits a nowhere-zero \(3\)-flow if \(\delta(G_1) \geq 2\) and \(G_2\) belongs to \(\mathcal{G}\), especially for the simplest case \(G_2 = K_2\). The main objective of this paper is to show that for any graph \(G\) with \(2 \leq \delta(G) \leq \Delta(G) \leq 3\), \(G \times K_2\) admits a nowhere-zero \(3\)-flow if and only if either every cycle in \(G\) contains an even number of vertices of degree \(2\) or every cycle in \(G\) contains an even number of vertices of degree \(3\). We also extend the sufficiency of this result to graphs \(G \times K_2\), where all odd vertices in \(G\) are of degree \(3\).

The notion of \(SDVFA\) (Strong Deterministic Variable Finite Automaton) of order \((s,t)\) was previously introduced by the author \([12]\). In this paper, we demonstrate the equivalence of \(SDVFA\) of order \((s,t)\) with DFA (Deterministic Finite Automaton), \(VDPA\) (Variable Deterministic Pushdown Automaton), NFA (Nondeterministic Finite Automaton), and \(\epsilon\)-NFA (extended Nondeterministic Finite Automaton). This equivalence is established by presenting conversions between \(SDVFA\) and \(DFA, VDFA, NFA\) (\(\epsilon\)-NFA), and vice versa.

Let \(G = (V, E)\) be a connected graph. \(G\) is \({super-\lambda}\) if every minimum edge cut of \(G\) isolates a vertex. Moreover, an edge set \(S \subseteq E\) is a \({restricted\; edge\; cut}\) of \(G\) if \(G – S\) is disconnected and every component of \(G – S\) has at least \(2\) vertices. The \({restricted \;edge\; connectivity}\) of \(G\), denoted by \(\lambda'(G)\), is the minimum cardinality of all restricted edge cuts. Let \(\xi(G) = \min\{d_G(u) + d_G(v) – 2: uv \in E(G)\}\). We say \(G\) is \({\lambda’-optimal}\) if \(\lambda'(G) = \xi(G)\). In this paper, we provide a sufficient condition for bipartite graphs to be both super-\(\lambda\) and \(\lambda’\)-optimal.

The thickness \(\theta(G)\) of a graph \(G\) is the minimum number of planar spanning subgraphs into which \(G\) can be decomposed. In this note, we determine the thickness of the complete tripartite graph \(K_{l,m,n}\) (\(1 \leq m \leq n\)) for the following cases: (1) \(l + m \leq 5\); (2) \(l + m\) is even and \(n > \frac{1}{2}(l + m – 2)\); (3) \(l + m\) is odd and \(n > (l + m – 2)(l + m – 1)\).

We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs.

Let \(D = (V, A)\) be a strongly connected digraph. \(D\) is called \(\lambda’\)-optimal if its restricted arc-connectivity equals the minimum arc degree. In this paper, we provide sufficient conditions for digraphs to be \(\lambda’\)-optimal.

In this paper, new families of Pell and Pell-Lucas numbers are introduced. In addition, we present the recurrence relations
and the generating functions of the new families for \(k = 2.\)

Consider a labeled and strongly oriented cycle \(\overrightarrow{C_m}\) and a set \(\mathcal{T} = \{\overrightarrow{C_n}, \overleftarrow{C_n}\}\), where \(\overrightarrow{C_n}\) and \(\overleftarrow{C_n}\) are two labeled and strongly oriented cycles with the same underlying graph and opposite orientations. Let \(h: E(\overrightarrow{C_m}) \to \Gamma\) be any function that sends every edge of \(\overrightarrow{C_m}\) to either \(\overrightarrow{C_n}\) or \(\overleftarrow{C_n}\). The primary goal of this paper is to study the underlying graph of the product \(\overrightarrow{C_m} \otimes_h \Gamma\), defined as follows:
\[ V(\overrightarrow{C_m} \otimes_h \Gamma) = V(\overrightarrow{C_m}) \times V(\overrightarrow{C_n}) \]
and
\[ ((a, b), (c, d)) \in E(\overrightarrow{C_m} \otimes_h \Gamma) \iff (a, c) \in E(\overrightarrow{C_m}) \wedge (b, d) \in h(a, c). \]
This product is of interest since it preserves various types of labelings, such as edge-magic and super edge-magic labelings. Additionally, we investigate the algorithmic complexity of determining whether a digraph \(\overrightarrow{D}\) can be factored using the product \(\otimes_h\), given a set of digraphs \(\Gamma\). This is the main topic of the third section of the paper.

Doubly resolvable \(2-(v,k,\lambda)\) designs \((DRDs)\) with small parameters and their resolutions which have orthogonal resolutions (\(RORs\)) are constructed and classified up to isomorphism. Exact values or lower bounds on the number of the nonisomorphic sets of \(7\) mutually orthogonal resolutions \((m-MORs)\) are presented. The implemented algorithms and the parameter range of this method are discussed, and the correctness of the computational results is checked in several ways.

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.

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.

For a graph \(G\) and a non-zero real number \(\alpha\), the graph invariant \(S_\alpha(G)\) is the sum of the \(\alpha^th\) power of the non-zero signless Laplacian eigenvalues of \(G\). In this paper, we obtain sharp bounds of \(S_\alpha(G)\) for a connected bipartite graph \(G\) on \(n\) vertices and a connected graph \(G\) on \(n\) vertices having a connectivity less than or equal to \(k\), respectively, and propose some open problems for future research.

In this paper we determine the scores of locally transitive tournaments and conversely, for such score we construct all locally transitive tournments having this score. This allows us to establish, for a given matrix, a test for the locally transitive property.

A graph is called a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. The direct product \(G \times H\) of graphs \(G\) and \(H\) has vertex set \(V(G) \times V(H)\) and edge set \(E(G \times H) = \{ (g_i, h_s)(g_j, h_t) \mid g_ig_j \in E(G) \text{ and } h_sh_t \in E(H) \}\). We prove that the direct product \(M_m(G) \times M_n(H)\) of the generalized Mycielskians of \(G\) and \(H\) is a cover graph if and only if \(G\) or \(H\) is bipartite.

For a primitive digraph \(D\) of order \(n\) and a positive integer \(m\) such that \(1 \leq m \leq n\), we define the \(m\)-competition index of \(D\), denoted by \(k_m(D)\), as the smallest positive integer \(k\) such that distinct vertices \(v_1, v_2, \ldots, v_m\) exist for each pair of vertices \(x\) and \(y\) with \(x \rightarrow^k v_i\) and \(y \rightarrow^k v_i\) for \(1 \leq i \leq m\) in \(D\). In this paper, we investigate the \(m\)-competition index of regular or almost regular tournaments.

A digraph \(D\) with \(e\) edges is labeled by assigning a distinct integer value \(\theta(v)\) from \(\{0, 1, \ldots, e\}\) to each vertex \(v\). The vertex values, in turn, induce a value \(\theta(u,v) = \theta(v) – \theta(u) \mod (e + 1)\) on each edge \((u,v)\). If the edge values are all distinct and nonzero, then the labeling is called a \emph{graceful labeling} of a digraph. Bloom and Hsu conjectured in 1985 that “all unicyclic wheels are graceful.” In this paper, we prove this conjecture.

Let \(m \geq 2\) be an integer and let \(G\) be a finite Abelian group of order \(p^n\), where \(p\) is an odd prime and \(n\) is a positive integer. In this paper, we derive necessary and sufficient conditions for the existence of an \(m\)-adic splitting of \(G\), and hence for the existence of polyadic codes (as ideals in an Abelian group algebra) of length \(p^n\). Additionally, we provide an algorithm to construct all \(m\)-adic splittings of \(G\). This work generalizes the results of Ling and Xing \([9]\) and Sharma, Bakshi, and Raka \([14]\).