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 \(0 \leq m \leq n(n-1)\).

A connected graph \(G\) is \({k-cycle \; resonant}\) if, for \(0 \leq t \leq k\), any \(t\) disjoint cycles \(C_1, C_2, \ldots, C_t\) in \(G\) imply a perfect matching in \(G – \bigcup_{i=1}^{t} V(C_i)\). \(G\) is \({cycle \; resonant}\) if it is \(k^*\)-cycle resonant, where \(k^*\) is the maximum number of disjoint cycles in \(G\). This paper proves that for outerplane graphs, \(2\)-cycle resonant is equivalent to cycle resonant and establishes a necessary and sufficient condition for an outerplanar graph to be cycle resonant. We also discuss the structure of \(2\)-connected cycle resonant outerplane graphs. Let \(\beta(G)\) denote the number of perfect matchings in \(G\). For any \(2\)-connected cycle resonant outerplane graph \(G\) with \(k\) chords, we show \(k+2 \leq \Phi(G) \leq 2^k + 1\) and provide extremal graphs for these inequalities.

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.

For a bipartite graph the extremal number for the existence of a specific odd (even) length path was determined in J. Graph Theory \(8 (1984), 83-95\). In this article, we conjecture that for a balanced bi-partite graph with partite sets of odd order the extremal number for an even order path guarantees many more paths of differing lengths.The conjecture is proved for a linear portion of the conjectured paths.

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.

Let \(K_{m} – H\) denote the graph obtained from the complete graph on \(m\) vertices, \(K_{m}\), by removing the edge set \(E(H)\) of \(H\), where \(H\) is a subgraph of \(K_{m}\). In this paper, we characterize the potentially \(K_{6} – 3K_{2}\)-graphic sequences, where \(pK_{2}\) is a matching consisting of \(p\) edges.

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.