Relations between the energy of graphs and other graph parameters

In this paper we give various relations between the energy of graphs and other graph parameters as Randić index, clique number, number of vertices and edges, maximum and minimum degree etc. Moreover, new bounds for the energy of complementary graphs are derived. Our results are based on the concept of vertex energy developed by G. Arizmendi and O. Arizmendi in [Lin. Algebra Appl. doi:10.1016/j.laa.2020.09.025].

In-situ Desaturation Tests by Electrolysis for Liquefaction Mitigation

Electrolytic desaturation is a potential method for improving the liquefaction resistance of the liquefiable foundation by reducing the soil saturation. In this study, in-situ desaturation tests were performed to investigate the resistivity of soil at different depth and the water level of the foundation under different current. The test results show that at constant currents of 1 A (Ampere, unit of the direct current), 2 A and 3 A, the saturation of the treated foundation reached 87%, 83% and 80%. During the electrolysis process, the generated gas migrates vertically and horizontally under the influence of buoyancy and gas pressure. In the end of electrolysis, the gas inside the sand foundation basically migrates vertically only. The higher current intensity employed for electrolysis will affect the uniformity and stability of the gas. At constant currents of 1 A, 2 A and 3 A, the difference between the maximum and minimum degree of saturation in the treated foundation was 14%, 18% and 19%; and after electrolysis halted for 144 h, the saturation in the treated foundation was 90%, 85% and 87%. The electricity consumption analysis indicates that the desaturation method has excellent economic benefits in the treatment of saturated sand foundations.

NEW UPPER BOUND ON THE LARGEST LAPLACIAN EIGENVALUE OF GRAPHS

Let G = (V;E) be a simple, undirected graph with maximum and minimum degree ∆ and respectively, and let A be the adjacency matrix and Q be the Laplacianmatrix of G. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several elds, such as randomized algorithms, combinatorial optimization problems and machine learning. In this paper, we compute lower and upper bounds for the largest Laplacian eigenvalue which is related with a given maximum and minimum degree and a given number of vertices and edges. We also compare our results in this paper with some known results.

Degree Kirchhoff Index of Bicyclic Graphs

Let G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

A note on the vertex-switching reconstruction

Bounds on the maximum and minimum degree of a graph establishing its reconstructibility from the vertex switching are given. It is also shown that any disconnected graph with at least five vertices is reconstructible.

Extreme degrees in random subgraphs of regular graphs

Let G(d) be a given simple d-regular graph on n labelled vertices, where dn is even. Such a graph will be called an initial graph. Denote by Gp(d) a random subgraph of G(d) obtained by removing edges, each with the same probability q — 1 —p, independently of all other edges (i.e. each edge remains in Gp(d) with probability p). In a recent paper [10] the asymptotic distributions of the number of vertices of a given degree in a random graph Gp(d) were given. The aim of this sequel is to present a wide variety of results devoted to probability distributions of the maximum and minimum degree of Gp(d) with respect to different values of the edge probability p and degree of regularity d. It should be noted here that very detailed results on a similar subject in the case when the initial graph is a complete graph (i.e. when d = n – 1) have already been obtained by Bollobás in the series of papers [2]–[4] (some additional information to the paper [4] was given in [9]). Also, in proving our results we will make use of some ideas given by Bollobás in these papers.