In this paper we define new types of generalizations in the distance sense of Lucas numbers. These generalizations are based on introduced recently the concept of \((2, k)\)-distance Fibonacci numbers.We study some properties of these numbers and present identities
which generalize known identities for Lucas numbers. Moreover, we show representations and interpretations of these numbers.

Let \(P\) be a planar point set with no three points collinear. A \(k\)-hole of \(P\) is a convex \(k\)-gon \(H\) such that the vertices of \(H\) are elements of \(P\) and no element of \(P\) lies inside \(H\). In this article, we prove that for any planar \(9\)-point set \(P\) with no three points collinear and at least \(5\) vertices on the boundary of the convex hull, \(P\) contains a \(5\)-hole and a disjoint \(3\)-hole.

This paper is an orthogonal continuation of the work of Belbachir and Belkhir in sense where we establish, using bijective proofs, recurrence relations and convolution identities between lines of \(r\)-Lah triangle. It is also established a symmetric function form for the \(r\)-Lah numbers.

In this paper, we investigate some properties of higher-order Cauchy of the second kind and poly-Cauchy of the second mixed type polynomials with umbral calculus viewpoint. From our investigation, we derive many interesting identities of higher-order Cauchy of the second kind and poly-Cauchy of the second kind mixed type polynomials.

The concept of signed cycle domination number of graphs, introduced by B. Xu [B. Xu, On signed cycle domination in graphs, Discrete Math. \(309 (2009)1007-1012]\), is extended to digraphs, denoted by \(\gamma’_{sc}(D)\) for a digraph \(D\). We establish bounds on \(\gamma_s(D)\), characterize all digraphs \(D\) with \(\gamma’_{sc}(D) = |A(D)|-2\), and determine the exact value of \(\gamma’_{sc}(D)\) for specific classes of digraphs \(D\). Furthermore, we define the parameter \(g'(m,n) = \min\{\gamma’_{sc}(D) \mid D \text{ is a digraph with } |V(D)| = n \text{ and } |A(D)| = m\}\) and obtain its value for all integers \(n\) and \(m\) satisfying \(0 \leq m \leq n(n-1)\).

In this paper we introduce a new kind of distance Pell numbers which are generated using the classical Fibonacci and Lucas numbers. Generalized companion Pell numbers is closely related to distance Pell numbers which were introduced in \([12]\). We present some relations between distance Pell numbers, distance companion Pell numbers and their connections with the Fibonacci numbers. To study properties of these numbers we describe their graph interpretations which in the special case gives a distance generalization of the Jacobsthal numbers. We also use the concept of a lexicographic product of graphs to obtain a new interpretation of distance Jacobsthal numbers.

Let \(D\) be a primitive digraph. Then there exists a nonnegative integer \(k\) such that there are walks of length \(k\) and \(k+1\) from \(u$ to \(v\) for some \(u,v \in V(D)\) (possibly \(u\) again ). Such smallest \(k\) is called the Lewin index of the digraph \(D\), denoted by \(l(D)\). In this paper, the extremal primitive digraphs with both Lewin index \(n — 2\) and girth \(2\) or \(3\) are determined.

In this paper, we investigate a generalized Catalan triangle defined by
\[\frac{k^m}{n} \binom{2n}{n-k}\]
for positive integers \(m\). We then compute weighted half binomial sums involving powers of generalized Fibonacci and Lucas numbers of the form
\[\sum\limits_{k=0}^{n} \binom{2n}{n+k} \frac{k^m}{n}X_{tk}^r,\]
where \(X_n\) either generalized Fibonacci or Lucas numbers, and \(t\) and \(r\) are integers, focusing on cases where \(1 \leq m \leq 6\). Furthermore, we outline a general methodology for computing these sums for larger values of \(m\).

A connected factor \(F\) of a graph \(G\) is a connected spanning subgraph of \(G\). If the degree of each vertex in \(F\) is an even number between \(2\) and \(2s\), where \(s\) is an integer, then \(F\) is a connected even \([2, 2s]\)-factor of \(G\). In this paper, we prove that every supereulerian \(K_{1,\ell+1},K_{1,\ell+1}+e\)-free graph (\(\ell \geq 2\)) contains a connected even \([2, 2\ell – 2]\)-factor.

In \([8]\), Weimin Li and Jianfei Chen studied split graphs such that the monoid of
all endomorphisms is regular. In this paper, we extend the study of \([11]\). We find
conditions such that regular endomorphism monoids of split graphs are completely
regular. Moreover, we find completely regular subsemigroups contained in the
monoid \(End(G)\).

A graph of order \(n\) is said to be \(k\)-factor-critical for non-negative integer \(k \leq n\) if the removal of any \(k\) vertices results in a graph with a perfect matching. For a \(k\)-factor-critical graph of order \(n\), it is called \({trivial}\) if \(k = n\) and \({non-trivial}\) otherwise. Since toroidal graphs are at most non-trivial \(5\)-factor-critical, this paper aims to characterize all non-trivial \(5\)-factor-critical graphs on the torus.

Let \(G\) be a simple graph of order \(n\) with \(\mu_1, \mu_2, \ldots, \mu_n\) as the roots of its matching polynomial. Recently, Gutman and Wagner defined the matching energy as \(\sum_{i=1}^{n} |\mu_i|\). In this paper, we first show that the Turán graph \(T_{r,n}\) is the \(r\)-partite graph of order \(n\) with maximum matching energy. Furthermore, we characterize the connected graphs (and bipartite graphs) of order \(n\) having minimum matching energy with \(m\) edges, where \(n+2 \leq m \leq 2n-4\) (and \(n \leq m\leq 2n-5\)).

The smallest bigraph that is edge-critical but not edge-minimal with respect to Hamilton laceability is the Franklin graph. Polygonal bigraphs\(^*\) \(P_{m,}\), which generalize one of the many symmetries of the Franklin graph, share this property of being edge-critical but not edge-minimal \([1]\). An enumeration of Hamilton paths in \(P_{m}\) for small \(m\) reveals surprising regularities: there are \(2^m\) Hamilton paths between every pair of adjacent vertices, \(3 \times 2^{m-2}\) between every vertex and a unique companion vertex, and \(3 \times 2^{m-2}\) between all other pairs. Notably, Hamilton laceability only requires at least one Hamilton path between every pair of vertices in different parts; remarkably, there are exponentially many.

In this paper, we develop an \(O(k^9 V^6)\) time algorithm to determine the cyclic edge connectivity of \(k\)-regular graphs of order \(V\) for \(k \geq 3\), which improves upon a previously known algorithm by Lou and Wang.

A graph \(G\) is called an \(L_1\)-graph if, for each triple of vertices \(u\), \(v\), and \(w\) with \(d(u,v) = 2\) and \(w \in N(u) \cap N(v)\), the condition \(d(u) + d(v) > |N(u) \cup N(v) \cup N(w)| – 1\) holds. This paper presents two results on the hamiltonicity of \(L_1\)-graphs.

Let \(S_n(k; |C_1|, \ldots, |C_k|)\) (\(k \geq 3\)) denote the \(n\)-vertex connected graph obtained from \(k\) cycles \(C_1, \ldots, C_k\) with a unique common vertex by attaching \(n – \sum_{i} |C_i|+k – 1\) pendent edges to it. In this paper, we show that among all \(n\)-vertex graphs with \(k\) edge-disjoint cycles, the following graphs have minimal Kirchhoff indices: (i) for \(n \leq 12\), \(S_7(3; 3,3, 3)\), \(S_8(3; 3,3, 4)\), \(S_9(3; 3, 4, 4)\), \(S_n(3; 4,4, 4)\) (\(n = 10, 11\)), \(S_{12}(3; 3, 3, 3)\), \(S_{12}(3; 3, 3, 4)\), \(S_{12}(3; 3, 4, 4)\), \(S_{12}(3; 4, 4, 4)\), \(S_9(4; 3, 3, 3, 3)\), \(S_{10}(4; 3, 3, 3, 4)\), \(S_{11}(4; 3, 3, 4, 4)\), \(S_{12}(4; 3, 3, 3, 3)\), \(S_{12}(4; 3, 3, 3, 4)\), \(S_{12}(4; 3, 3, 4, 4)\), \(S_{12}(4; 3, 4, 4, 4)\), \(S_{11}(5; 3, 3, 3, 3, 3)\), \(S_{12}(5; 3, 3, 3, 3, 3)\), \(S_{12}(5; 3, 3, 3, 3, 4)\); (ii) for \(n > 12\), \(S_n(k; 3, \ldots, 3)\). Additionally, we obtain lower bounds for the Kirchhoff index of \(n\)-vertex graphs with \(k\) edge-disjoint cycles.

We investigate the conditions under which an association scheme exists on the set of lines of a regular near hexagon with quads of order \((s, t_2)\) passing through every two points at distance \(2\). Specifically, we determine all regular near hexagons admitting such an association scheme when \(s \geq t_2\), while the case \(t^2 > s\) remains open.

Let \(G\) be a connected graph of order \(n\) with Laplacian eigenvalues \(\mu_1 \geq \mu_2 \geq \cdots \geq \mu_n = 0\). The Laplacian-energy-like invariant (\(LEL\) for short) of \(G\) is defined as \(\text{LEL} = \sum_{i=1}^{n-1} \sqrt{\mu_i}\). In this paper, we investigate the asymptotic behavior of the \(LEL\) of iterated line graphs of regular graphs. Furthermore, we derive the exact formula and asymptotic formula for the \(LEL\) of square, hexagonal, and triangular lattices with toroidal boundary conditions.

Let \(S_{r,l}\) be a generalized star on \(rl+1\) vertices with central vertex \(v\). Let \(H_v\) be a graph of order \(m\) with a specified vertex \(v\) of degree \(m-1\). For simple connected graphs \(G_{r,l,H_v}\), obtained by attaching \(v\) of \(H_v\) to each vertex of \(S_{r,l}\) except the central vertex, we derive the adjacency, Laplacian, and signless Laplacian spectrum of \(G_{r,l,H_v}\) in terms of the corresponding spectrum of \(S_{r,l}\) and \(H_v\). Furthermore, we extend these results to obtain the adjacency, Laplacian, and signless Laplacian characteristic polynomials of general graphs.

An important invariant of an interconnection network is its surface area, the number of nodes at distance \(i\) from a node. We derive explicit formulas, via two different approaches: direct counting and generating function, for the surface areas of the alternating group graph and the split-star graph, two Cayley graphs that have been
proposed to interconnect processors in a parallel computer.

This paper is based on the splitting operation for binary metroids that was introduced by Raghunathan, Shikare, and Waphare [Discrete Math. \(184 (1998), p.267-271\)] as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which cographic matroids \(M\) have the property that the splitting operation, by every pair of elements,on \(M\) yields a cographic matroid. This problem is solved by proving that there are exactly five minor-minimal matroids that do not have this property.

In 2003, Li introduced the concept of implicit weighted degree, denoted by \(id^w(v)\) for a vertex \(v\) in a weighted graph. In this paper, we prove that: Let \(G\) be a 2-connected weighted graph satisfying: (a) the implicit weighted degree sum of any three independent vertices is at least \(m\); (b) for each induced claw, modified claw, and FP, all edges have the same weight. Then \(G\) contains either a hamiltonian cycle or a cycle of weight at least \(\frac{2}{3}m\).

The Fibonacci \((p, r)\)-cube is an interconnection topology that unifies various connection topologies, including the hypercube, classical Fibonacci cube, and postal network. While classical Fibonacci cubes are known to be partial cubes, we demonstrate that a Fibonacci \((p, r)\)-cube is a partial cube if and only if either \(p = 1\) or \(p \geq 2\) and \(r \leq p + 1\). Furthermore, we establish that for Fibonacci \((p, r)\)-cubes, the properties of being almost-median graphs, semi-median graphs, and partial cubes are equivalent.

In this paper, we establish the equivalence between semi-
deterministic virtual finite automaton\((SDVFA)\) of order \((s,t)\) and
and regular grammar.

For a graph \(G\), a \({trail}\) is a vertex-edge alternating sequence \(v_0, e_1, v_1, e_2, \ldots, e_{k-1},v_{k-1}, e_k, v_k\) such that all \(e_i\)’s are distinct and \(e_i = v_{i-1}v_i\) for all \(i\). For \(u, v \in V(G)\), a \((u,v)\)-trail of \(G\) is a trail in \(G\) originating at \(u\) and terminating at \(v\). A closed trail is a \((u,v)\)-trail with \(u = v\). A trail \(H\) is a spanning trail of \(G\) if \(V(H) = V(G)\). Let \(X \subseteq E(G)\) and \(Y \subseteq E(G)\) with \(X \cap Y = \emptyset\). This paper studies the minimum edge-connectivity of \(G\) such that for any \(u, v \in V(G)\) (including \(u = v\)), \(G\) has a spanning \((u, v)\)-trail \(H\) with \(X \subseteq E(H)\) and \(Y \cap E(H) = \emptyset\).

In this paper we settle a long-standing open problem. We prove that it
is \(NP\)-hard to recognize \(T\)-tenacious graphs for any fixed positive rational
number \(T\)

In this paper, we deal with the transitive relations on a finite $n$-element set. The transitive relations are interpreted as Boolean matrices. A special class of transitive relations are constructed and enumerated, which can generate all transitive
relations on a finite n-element set by intersection operation. Besides, several necessary and sufficient conditions that a relation
\(R\) is transitive are given.

In this paper, we obtain an upper bound on the order of a blockwise-burst \([11]\) that can be detected by a row-cyclic array code \([10]\) and obtain the fraction of blockwise-bursts of order exceeding the upper bound that go undetected. We also give a decoding algorithm for the correction of blockwise-bursts in row-cyclic array codes.

In this paper we study defensive alliances in some specific regular graphs, the circulant graphs, i.e. Cayley graphs on a cyclic group.The critical defensive alliances of a circulant graph of degree at most \(6\) are completely determined. For the general case, we give tight lower and upper bounds on the alliance number of a circulant graph with \(d\) generators.

The maximum number of non-isomorphic one-edge extensions \(M(t, n, f)\) of a graph of size \(t\), order \(n\), and vertex degree bounded by \(f\) for \(3 \leq f \leq n-2\) is considered. An upper bound for \(M(t, n, f)\) is obtained, and for the case \(f = n-2\), the exact value is given. Tables are provided for all values of \(M(t, n, f)\) for up to \(n = 12\), \(\left\lfloor \frac{f-1}{2} \right\rfloor < t \leq \left\lfloor \frac{nf}{2} \right\rfloor\), and \(3 \leq f \leq n-2\). Additionally, the relation of these results to the transition digraph for the Random \(f\)-Graph Process, a Markov process concerning graphs with vertex degree bounded by \(f\), is noted.

In this paper, we characterize all spanning trees of the \(r\)-cyclic graph \(G_{n,r}\). We provide the formulation of \(f\)-vectors associated with spanning simplicial complexes \(\Delta_s(G_{n,r})\) and, consequently, deduce a formula for computing the Hilbert series of the Stanley-Reisner ring \(k[\Delta_s(G_{n,r})]\). For the facet ideal \(I(\Delta(G_{n,r}))\), we characterize all associated primes. Specifically, for uni-cyclic graphs with cycle length \(m_i\), we prove that the facet ideal \(I(\Delta(G_{n,1}))\) has linear quotients with respect to its generating set. Furthermore, we establish that projdim \((I_{\mathcal{F}}(\Delta_s(G_{n,1}))) = 1\) and \(\beta_i(I_{\mathcal{F}}(\Delta_(G_s{n,1}))) = m_i\) for \(i \leq 1\).

We consider the one-player game called Dundee, where a deck consists of \(s_i\) cards of value \(i\), for \(i = 1, \ldots, v\), and an integer \(m \leq s_1 + \cdots + s_v\). Over \(m\) rounds, the player names a number between \(1\) and \(v\) and draws a random card from the deck, losing if the named number matches the drawn value in at least one round. The famous Problem of Thirteen, proposed by Montmort in 1708, asks for the winning probability when \(v = 13\), \(s_1 = \cdots = s_{13} = 4\), \(m = 13\), and the player names the sequence \(1, \ldots, 13\). Studied by mathematicians including J. and N. Bernoulli, De Moivre, Euler, and Catalan, this problem’s strategic aspects remain unexplored. We investigate two variants: one where the player’s Round \(i\) bid depends on previous rounds’ drawn values, which we completely solve, and another where the player must specify all \(m\) bids in advance, solving this for \(s_1 = \cdots = s_v\) and arbitrary \(m\).

Let \(n\) be a positive integer with \(n\geq 2\) and \([n] := \{1, 2, \ldots, n\}\). An \(m\)-partial injective map of \([n]\) is a pair \((A, f)\), where \(A\) is an \(m\)-subset of \([n]\) and \(f: A \rightarrow [n]\) is an injective map. Let \(P =L \cup \{I\}\), where \(L\) is the set of all partial injective maps of \([n]\). Partially ordering \(P\) by ordinary or reverse inclusion yields two families of finite posets. This article proves that these posets are atomic lattices, discusses their geometricity, and computes their characteristic polynomials.

In this paper we study defensive alliances in some regular graphs. We determine which subgraphs could a critical defensive alliance of a graph \(G\) induce, if \(G\) is \(6\)-regular and the cardinality of the alliance is at most \(8\).

We present mean and non mean graphs of order \(\leq 6\), and give an upper bound for the number of edges of a graph with certain number of vertices to be a mean graph, and we show that the maximum vertex degree could be found in mean graphs depending on the number of edges. Also, we construct families of mean graphs depending on other mean and non mean graphs.

Let \(G = (V, E)\) be a finite, simple, and undirected graph of order \(p\) and size \(q\). A super edge-magic total labeling of a graph \(G\) is a bijection \(\lambda: V(G) \cup E(G) \rightarrow \{1, 2, \ldots, p + q\}\), where vertices are labeled with \(1, 2, \ldots, p\) and there exists a constant \(t\) such that \(f(x) + f(xy) +f(y) = t\), for every edge \(xy \in E(G)\). The super edge-magic deficiency of a graph \(G\), denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) such that \(G \cup nK_1\) has a super edge-magic total labeling, or \(\infty\) if no such \(n\) exists. In this paper, we investigate the super edge-magic deficiency of a forest consisting of stars.