Mathematics Interdisciplinary Research
Volume:4 Issue: 2, Autumn 2019

‎In addition to his countless contributions to spectral graph theory‎, some works of Dragos Cvetkovic are concerned with chemical problems‎. These are briefly outlined‎, ‎with emphasis on his collaboration with‎ the present author‎.

In the paper we consider a generalized join operation, that is, the H-join on graphs where H is an arbitrary graph. In terms of Seidel matrix of graphs we determine the Seidel spectrum of the graphs obtained by this operation on regular graphs. Some additional consequences regarding S-integral complete split graphs are also obtained, which allows to exhibit many infinite families of Seidel integral complete split graphs.

The graph energy is the sum of absolute values of the eigenvalues of the (0, 1)-adjacency matrix. Oboudi recently obtained lower bounds for graph energy, depending on the largest and smallest graph eigenvalue. In this paper, a few more Oboudi-type bounds are deduced.

The PageRank (PR) algorithm is the base of Google search engine. In this paper, we study the PageRank sequence for undirected graphs of order six by PR vector. Then, we provide an ordering for graphs by variance of PR vector which it’s variation is proportional with variance of degree sequence. Finally, we introduce a relation between domination number and PR-variance of graphs.

In this paper, for a connected graph G and a real alpha (not equal to) 0, we define a new graph invariant sigma_alpha (G)-as the sum of the alphath powers of the normalized signless Laplacian eigenvalues of G. Note that sigma_1/2 (G) is equal to Randic (normalized) incidence energy which have been recently studied in the literature [5, 15]. We present some bounds on sigma_alpha(G) (alpha (not equal to) 0, 1) and also consider the special case alpha = 1/2.

Abstract. The Laplacian characteristic polynomial of an n-vertex graph G has the form f(G,x) = x^n+∑l_ix^n-i. In this paper, the fourth and fifth coefficient of f(G,x), will be investigated, where G is a T(k,t) tree in which a rooted tree with degree sequence k,k,···,k,1,1,···,1 is denoted by T(k,t).

In this paper, we present upper bounds for the multiplicative forgotten topological index of several graph operations such as sum, Cartesian product, corona product, composition, strong product, disjunction and symmetric difference in terms of the F–index and the first Zagreb index of their components. Also, we give explicit formulas for this new graph invariant under two graph operations such as union and Tensor product. Moreover, we obtain the expressions for this new graph invariant of subdivision graphs and vertex – semitotal graphs. Finally, we compare the discriminating ability of indices.

In this paper we investigate the geometric structures of M(n, 2) containing n points in R^3 having two distinct distances. We will show that up to pseudo-equivalence there are 5 constructible models for M(4, 2) and 17 constructible models for M(5, 2).

In this paper we present a generalization of the aforementioned bound for all trees in terms of the order and maximum degree. We also give a lower bound on the second Zagreb coindex of trees.

The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. In this paper we study the distinguishing number and the distinguishing index of the join of two graphs G and H, i.e., G+H. We prove that 0≤ D(G+H)-max{D(G),D(H)}≤ z, where z depends on the number of some induced subgraphs generated by some suitable partitions of V(G) and V(H). Let Gk be the k-th power of G with respect to the join product. We prove that if $G$ is a connected graph of order n ≥ 2, then Gk has the distinguishing index 2, except D'(K_2+K_2)=3.

‎The aim of this paper is to introduce some results for the F-index of the tree structures without any information on the exact values of vertex degrees‎. Three martingales related to the first Zagreb index and F-index are given‎.

By using bifurcation theory of planar dynamical systems, we classify all bounded travelling wave solutions of the general Burgers-Boussinesq equation, and we give their corresponding phase portraits. In different parametric regions, different types of trav- elling wave solutions such as solitary wave solutions, cusp solitary wave solutions, kink(anti kink) wave solutions and periodic wave solutions are simulated. Also in each parameter bifurcation sets, we obtain the exact explicit parametric representation of all travelling wave solutions.

‎The deleted lexicographic‎ ‎product G[H]-nG of graphs G and H is a graph with vertex set V(G)×V(H)‎ and u=(u1‎, ‎v1) is adjacent with v=(u2‎, ‎v2) whenever (u1=u2 and‎ v1 is adjacent with v2) or (v1 ≠ v2 and u1 is adjacent with u2)‎. ‎In this paper‎, ‎we compute the exact values of the Wiener‎, ‎vertex PI and Zagreb indices‎ of deleted lexicographic product of graphs‎. ‎Applications of our results under some examples are presented‎.

In this paper, we aim to introduce Ciric type G-contractions using directed graphs in metric spaces and then to investigate the existence and uniqueness of best proximity points for them. We also discuss the main theorem and list some consequences of it.

For any nonempty set Ω and k-subset Λ, the k-intersection graph, denoted by Γm(Ω,Λ), is an undirected simple graph whose vertices are all m-subsets of Ω and two distinct vertices A and B are adjacent if and only if A∩B ⊈ Λ. In this paper, we determine diameter, girth, some numerical invariants and planarity, Hamiltonian and perfect matching of these graphs. finally we investigate their adjacency matrices.

Let λ1(G), λ2(G),..., λs(G) be the distinct eigenvalues of G with multiplicities t1, t2,..., ts, respectively. The multiset {λ1(G)t1, λ2(G)t2,..., λs(G)ts} of eigenvalues of A(G) is called the spectrum of G. For two graphs G and H, if their spectrum are the same, then G and H are said to be co-spectral. The aim of this paper is to determine co-spectral permutation graphs with respect to automorphism group of graph G.