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.

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.