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 \(i \leq 1\).