Construction of L-Borderenergetic Graphs

If a graph G of order n has the Laplacian energy same as that of complete graph Kn then G is said to be L-borderenergeic graph. It is interesting and challenging as well to identify the graphs which are L-borderenergetic as only few graphs are known to be L-borderenergetic. In the present work we have investigated a sequence of L-borderenergetic graphs and also devise a procedure to find sequence of L-borderenergetic graphs from the known L-borderenergetic graph.

On graphs with minimal distance signless Laplacian energy

For a simple connected graph G of order n having distance signless Laplacian eigenvalues ρ 1 Q ≥ ρ 2 Q ≥ ⋯ ≥ ρ n Q \rho _1^Q \ge \rho _2^Q \ge \cdots \ge \rho _n^Q , the distance signless Laplacian energy DSLE(G) is defined as D S L E ( G ) = ∑ i = 1 n | ρ i Q - 2 W ( G ) n | DSLE\left( G \right) = \sum\nolimits_{i = 1}^n {\left| {\rho _i^Q - {{2W\left( G \right)} \over n}} \right|} where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph Kk ∨ ( Kt∪ Kn−k−t), 1 ≤ t ≤ ⌊ n - k 2 ⌋ 1 \le t \le \left\lfloor {{{n - k} \over 2}} \right\rfloor has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.

On Harary energy and Reciprocal distance Laplacian energies1

Let G be an graph simple, undirected, connected and unweighted graphs. The Reciprocal distance energy of a graph G is equal to the sum of the absolute values of the reciprocal distance eigenvalues. In this work, we find a lower bound for the Harary energy, reciprocal distance Laplacian energy and reciprocal distance signless Laplacian energy of a graph. Moreover, we find relationship between the Harary energy and Reciprocal distance Laplacian energies.

On Laplacian Equienergetic Signed Graphs

The Laplacian energy of a signed graph is defined as the sum of the distance of its Laplacian eigenvalues from its average degree. Two signed graphs of the same order are said to be Laplacian equienergetic if their Laplacian energies are equal. In this paper, we present several infinite families of Laplacian equienergetic signed graphs.

The minimum vertex–vertex dominating Laplacian energy of a graph

Let [Formula: see text] denote the set of all blocks of a graph [Formula: see text]. Two vertices are said to vv-dominate each other if they are vertices of the same block. A set [Formula: see text] is said to be vertex–vertex dominating set (vv-dominating set) if every vertex in [Formula: see text] is vv-dominated by some vertex in [Formula: see text]. The vv-domination number [Formula: see text] is the cardinality of the minimum vv-dominating set of [Formula: see text]. In this paper, we introduce new kind of graph energy, the minimum vv-dominating Laplacian energy of a graph denoting it as LE[Formula: see text]. It depends both on the underlying graph of [Formula: see text] and the particular minimum vv-dominating set of [Formula: see text]. Upper and lower bounds for LE[Formula: see text] are established and we also obtain the minimum vv-dominating Laplacian energy of some family of graphs.

On a conjecture of Laplacian energy of trees

Let [Formula: see text] be a simple graph with [Formula: see text] vertices, [Formula: see text] edges having Laplacian eigenvalues [Formula: see text]. The Laplacian energy LE[Formula: see text] is defined as LE[Formula: see text], where [Formula: see text] is the average degree of [Formula: see text]. Radenković and Gutman conjectured that among all trees of order [Formula: see text], the path graph [Formula: see text] has the smallest Laplacian energy. Let [Formula: see text] be the family of trees of order [Formula: see text] having diameter [Formula: see text]. In this paper, we show that Laplacian energy of any tree [Formula: see text] is greater than the Laplacian energy of [Formula: see text], thereby proving the conjecture for all trees of diameter [Formula: see text]. We also show the truth of conjecture for all trees with number of non-pendent vertices at most [Formula: see text]. Further, we give some sufficient conditions for the conjecture to hold for a tree of order [Formula: see text].

Novel Concept of Energy in Bipolar Single-Valued Neutrosophic Graphs with Applications

The energy of a graph is defined as the sum of the absolute values of its eigenvalues. Recently, there has been a lot of interest in graph energy research. Previous literature has suggested integrating energy, Laplacian energy, and signless Laplacian energy with single-valued neutrosophic graphs (SVNGs). This integration is used to solve problems that are characterized by indeterminate and inconsistent information. However, when the information is endowed with both positive and negative uncertainty, then bipolar single-valued neutrosophic sets (BSVNs) constitute an appropriate knowledge representation of this framework. A BSVNs is a generalized bipolar fuzzy structure that deals with positive and negative uncertainty in real-life problems with a larger domain. In contrast to the previous study, which directly used truth and indeterminate and false membership, this paper proposes integrating energy, Laplacian energy, and signless Laplacian energy with BSVNs to graph structure considering the positive and negative membership degree to greatly improve decisions in certain problems. Moreover, this paper intends to elaborate on characteristics of eigenvalues, upper and lower bound of energy, Laplacian energy, and signless Laplacian energy. We introduced the concept of a bipolar single-valued neutrosophic graph (BSVNG) for an energy graph and discussed its relevant ideas with the help of examples. Furthermore, the significance of using bipolar concepts over non-bipolar concepts is compared numerically. Finally, the application of energy, Laplacian energy, and signless Laplacian energy in BSVNG are demonstrated in selecting renewable energy sources, while optimal selection is suggested to illustrate the proposed method. This indicates the usefulness and practicality of this proposed approach in real life.