In this paper, the relations of maximum degree energy and maximum reserve degree energy of a complete graph after removing a vertex have been shown to be proportional to the energy of the complete graph. The results of splitting the graph and shadow graphs are also presented for the complete graph after removing a vertex.

Based on the Hermitian adjacency matrices of second kind introduced by Mohar [1] and weighted adjacency matrices introduced in [2], we define a kind of index weighted Hermitian adjacency matrices of mixed graphs. In this paper we characterize the structure of mixed graphs which are cospectral to their underlying graphs, then we determine a upper bound on the spectral radius of mixed graphs with maximum degree \(\Delta\), and characterize the corresponding extremal graphs.

Modified group divisible designs MGD\((k, \lambda, m, n)\) are extensively studied because of an intriguing combinatorial structure that they possess and their applications. In this paper, we present a generalization of MGDs called GMGD\((k, \lambda_1, \lambda_2, m, n)\), and we provide some elementary results and constructions of some special cases of GMGDs. In addition, we show that the necessary conditions are sufficient for the existence of a GMGD\((3, \lambda, 2\lambda, m, n)\) for any positive integer \(\lambda\), and a GMGD\((3, 2, 3, m, n)\). Though not a general result, the construction of a GMGD\((3, 3, 2, 2, 6)\) given in the paper is worth mentioning in the abstract. Along with another example of a GMGD\((3, 3, 2, 2, 4)\), and \(n\) to \(tn\) construction, we have families of GMGD\((3, 3\lambda, 2\lambda, 2, n)\)s for \(n = 4t\) or \(6t\) when \(t \equiv 0, 1 \pmod 3\), for any positive integer \(\lambda\).

We show that connected, bicyclic graphs on nine edges with at least one cycle other than \(C_3\) decompose the complete graphs \(K_{18k}\) and \(K_{18k+1}\), for \(k\geq1\), when the necessary conditions allow for such a decomposition. This complements previous results by Freyberg, Froncek, Jeffries, Jensen, and Sailstad on connected bicyclic triangular graphs.

In the realm of graph theory, recent developments have introduced novel concepts, notably the \(\nu\varepsilon\)-degree and \(\varepsilon\nu\)-degree, offering expedited computations compared to traditional degree-based topological indices (TIs). These TIs serve as indispensable molecular descriptors for assessing chemical compound characteristics. This manuscript aims to meticulously compute a spectrum of TIs for silicon carbide \(SiC_{4}\)-\(I[r,s]\), with a specific focus on the \(\varepsilon\nu\)-degree Zagreb index, the \(\nu\varepsilon\)-degree Geometric-Arithmetic index, the \(\varepsilon\nu\)-degree Randić index, the \(\nu\varepsilon\)-degree Atom-bond connectivity index, the \(\nu\varepsilon\)-degree Harmonic index, and the \(\nu\varepsilon\)-degree Sum connectivity index. This study contributes to the ongoing advancement of graph theory applications in chemical compound analysis, elucidating the nuanced structural properties inherent in silicon carbide molecules.

Graph theory has experienced notable growth due to its foundational role in applied mathematics and computer science, influencing fields like combinatorial optimization, biochemistry, physics, electrical engineering (particularly in communication networks and coding theory), and operational research (with scheduling applications). This paper focuses on computing topological properties, especially in molecular structures, with a specific emphasis on the nanotube \(HAC_{5}C_{7}[w,t]\).

Let \(\beta_{H}\) denote the orbit graph of a finite group \(H\). Let \(\zeta\) be the set of commuting elements in \(H\) with order two. An orbit graph is a simple undirected graph where non-central orbits are represented as vertices in \(\zeta\), and two vertices in \(\zeta\) are connected by an edge if they are conjugate. In this article, we explore the Laplacian energy and signless Laplacian energy of orbit graphs associated with dihedral groups of order $2w$ and quaternion groups of order \(2^{w}\).

In this paper, we introduce the concept of the generalized \(3\)-rainbow dominating function of a graph \(G\). This function assigns an arbitrary subset of three colors to each vertex of the graph with the condition that every vertex (including its neighbors) must have access to all three colors within its closed neighborhood. The minimum sum of assigned colors over all vertices of G is defined as the \(g_{3}\)-rainbow domination number, denoted by \(\gamma_{g3r}\). We present a linear-time algorithm to determine a minimum generalized 3-rainbow dominating set for several graph classes: trees, paths \((P_n)\), cycles \((C_n)\), stars (\(K_1,n)\), generalized Petersen graphs \((GP(n,2)\), GP \((n,3))\), and honeycomb networks \((HC(n))\).

A module \(M\) over a commutative ring is termed an \(SCDF\)-module if every Dedekind finite object in \(\sigma[M]\) is finitely cogenerated. Utilizing this concept, we explore several properties and characterize various types of \(SCDF\)-modules. These include local \(SCDF\)-modules, finitely generated $SCDF$-modules, and hollow \(SCDF\)-modules with \(Rad(M) = 0 \neq M\). Additionally, we examine \(QF\) SCDF-modules in the context of duo-ri

Let \(G=(V(G),E(G))\) be a graph with \(p\) vertices and \(q\) edges. A graph \(G\) of size \(q\) is said to be odd graceful if there exists an injection \(\lambda: V(G) \to {0,1,2,\ldots,2q-1}\) such that assigning each edge \(xy\) the label or weight \(|\lambda(x) – \lambda(y)|\) results in the set of edge labels being \({1,3,5,\ldots,2q-1}\). This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the \(W\)-tree, denoted as \(WT(n,k)\).