Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The multiplicity of Laplacian eigenvalue two in a connected graph with a perfect matching

The multiplicity of Laplacian eigenvalue two in unicyclic graphs

Let G be a connected graph. The Szeged index of G is defined as $$Sz(G)=\sum \nolimits _{e=uv\in E(G)}n_{u}(e|G)n_{v}(e|G)$$ , where $$n_{u}(e|G)$$ (resp., $$n_{v}(e|G)$$ ) is the number of vertices whose distance to vertex u (resp., v) is smaller than the distance to vertex v (resp., u), and $$n_{0}(e|G)$$ is the number of vertices equidistant from both ends of e. Let $$\mathcal {M}(2\beta )$$ be the set of unicyclic graphs with order $$2\beta $$ and a perfect matching. In this paper, we determine the minimum value of Szeged index and characterize the extremal graph with the minimum Szeged index among all unicyclic graphs with perfect matchings.

Spectra of copies of a generalized Bethe tree attached to any graph

‎For a simple graph G‎, ‎the signless Laplacian Estrada index is defined as SLEE(G)=∑ni=1eqi‎, ‎where q1‎, ‎q2‎,...‎, ‎qn are the eigenvalues of the signless Laplacian matrix of G‎. ‎In this paper‎, ‎we first characterize the unicyclic graphs with the first two largest and smallest SLEE's and then determine the unique unicyclic graph with maximum SLEE among all ‎unicyclic graphs on n vertices with a given diameter‎. ‎All extremal graphs‎, ‎which have been introduced in our results are also extremal with respect to the signless Laplacian ‎resolvent energy‎.

The Symmetric division deg (SDD) index is a well-established valuable index in the analysis of quantitative structure-property and structure-activity relationships for molecular graphs. In this paper, we study the range of SDD-index for special classes of trees and unicyclic graphs. We present the first four lower bounds for SDD-index of trees and unicyclic graphs, which admit a perfect matching and find the subclasses of graphs that attain these bounds. Further, we also compute the upper bounds of SDD-index for the collection of molecular graphs, namely the trees and unicyclic graphs, each having maximum degree four and that admit a perfect matching.

Let be the characteristic polynomial of the signless Laplacian matrix of a simple graph of order , where and are the degree diagonal and adjacency matrices of , respectively. In this paper, we focus on how the signless Laplacian coefficients of unicyclic graphs change after some graph transformations. These results can be used to characterize all extremal unicyclic graphs having the minimal signless Laplacian coefficients in the set of all unicyclic graphs of order and the matching number . Moreover, the unicyclic graphs with minimum incidence energy in are also characterized.

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ r (G) ≤ μ(G), where μ r (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ r (G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.

The first general multiplicative Zagreb index of a graph G is defined as $$P_1^a (G) = \prod _{v \in V(G)} (deg_G (v))^a$$ and the second general multiplicative Zagreb index is $$P_2^a (G) = \prod _{v \in V(G)} (deg_G (v))^{a \, deg_G (v)}$$ , where V(G) is the vertex set of G, $$deg_{G} (v)$$ is the degree of v in G and $$a \ne 0$$ is a real number. We present lower and upper bounds on the general multiplicative Zagreb indices for trees and unicyclic graphs of given order with a perfect matching. We also obtain lower and upper bounds for trees and unicyclic graphs of given order and matching number. All the trees and unicyclic graphs which achieve the bounds are presented, thus our bounds are sharp. Bounds for the classical multiplicative Zagreb indices are special cases of our theorems and those bounds are new results as well.

The revised edge-Szeged index of a connected graph $G$ is defined as Sze*(G)=∑e=uv∊E(G)( (mu(e|G)+(m0(e|G)/2)(mv(e|G)+(m0(e|G)/2) ), where mu(e|G), mv(e|G) and m0(e|G) are, respectively, the number of edges of G lying closer to vertex u than to vertex v, the number of edges of G lying closer to vertex v than to vertex u, and the number of edges equidistant to u and v. In this paper, we give an effective method for computing the revised edge-Szeged index of unicyclic graphs and using this result we identify the minimum revised edge-Szeged index of conjugated unicyclic graphs (i.e., unicyclic graphs with a perfect matching). We also give a method of calculating revised edge-Szeged index of the joint graph.

The harmonic index of a graph G is defined as the sum of weights 2/ d(u)+d(v) of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G, respectively. In this paper, we determine the graph with minimum harmonic index among all unicyclic graphs with a perfect matching. Moreover, the graph with minimum harmonic index among all unicyclic graphs with a given matching number is also determined.

The Randić index R(G) of a graph G is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given size of matching.

Some results on the signless Laplacians of graphs

Bounds for the signless Laplacian energy

We investigate graphs whose signless Laplacian matrix has three distinct eigenvalues. We show that the largest signless Laplacian eigenvalue of a connected graph G with three distinct signless Laplacian eigenvalues is noninteger if and only if G = K n − e for n ≥ 4, where K n − e is the n vertex complete graph with an edge removed. Moreover, examples of such graphs are given in this article.