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Omega polynomial was introduced by Diudea. Following Omega polynomial, the Sadhana polynomial was defined by Ashrafi et al. In this paper we compute Omega and Sadhana polynomials of three classes of dendrimers  

The ABC index is a topological index was defined as where dG(u) denotes degree of vertex u. Now we define a new version of ABC index as where The goal of this paper is further the study of the ABC4 index  

The Zagreb indices have been introduced more than thirty years ago by Gutman and Trinajstić. In this paper we introduce a new version of Zagreb indices and then we compute them for an infinite family of nanostar dendrimers  

Let ∑ be the class of finite graphs. A topological index is a function Top from ∑ into real numbers with this property that Top(G) = Top(H), if G and H are isomorphic. is defined as half sum of the distances between all the pairs of vertices in a molecular graph. The goal of this paper is to further the study of Wiener index of special case of a chain of fullerenes  

The energy ϵ(G) of a graph G is the sum of the absolute values of all eigenvalues of G. Zhou in (MATCH Commun. Math. Comput. Chem. 55 (2006) 91-94) studied the problem of bounding the graph energy in terms of the minimum degree together with other parameters. He proved his result for quadrangle-free graphs. Recently, in (MATCH Commun. Math. Comput. Chem. 81 (2019) 393-404) it is shown that for every graph G, ϵ(G) ≥ 2δ(G), where δ(G) is the minimum degree of G, and the equality holds if and only if G is a complete multipartite graph with equal size of parts. Here, we provide a short proof for this result. Also, we give an affirmative answer to a problem proposed in (MATCH Commun. Math.... 

Two edges e and f of a plane graph G are in relation opposite, e op f, if they are opposite edges of an inner face of G. Relation op enables the partition the edge set of G into opposite edge strips ops. On this ground, Diudea defined Omega and Theta polynomial while Ashrafi et al. defined the Sadhana polynomial. In this paper a weighted version of these polynomials was introduced and several relations between them are demonstrated. Some molecular weights are suggested in view of using the derived topological indices in correlational studies  

Some new relations have been established between Wiener indices, stability numbers and clique numbers for several classes of composite graphs that arise via graph products. For three of considered operations we show that they make a multiplicative pair with the clique number  

A new counting polynomial, called Omega, was recently proposed by Diudea. It is defined on the ground of opposite edge strips "ops". The Sadhana polynomial has been defined to evaluate the Sadhana index of a molecular graph. A third one, called Theta polynomial is also derived from ops. In this paper we compute Omega, Sadhana and Theta polynomials of the Cartesian product graphs  

A new counting polynomial, called Omega Ω(G,x), was recently proposed by Diudea. It is defined on the ground of "opposite edge strips" ops. Two related polynomials: Sadhana Sd(G,x) and Theta Θ(G,x) polynomials can also be calculated by ops counting. Close formulas for calculating these three polynomials in infinite nano-structures resulted by embedding the titanium dioxide pattern in plane, cylinder and torus are derived. For the design of titanium dioxide pattern, a procedure based on a sequence of map operations is proposed  

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ(G)≥3, then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D1 is a primitive regular orientation and D2 is a non-regular orientation of a given graph, then for sufficiently large t, the number of closed walks of length t in D1 is more than the number of closed walks of length t in D2. © 2016 Elsevier Inc  

The singular values of a matrix A are defined as the square roots of the eigenvalues of A⁎A, and the energy of A denoted by E(A) is the sum of its singular values. The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A=(BDD⁎C), then E(A)≥2E(D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E(H) is an edge cut of G, then E(H)≤E(G), i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known... 

The modified eccentricity connectivity polynomial of a connected graph G is defined as Ξ(G; x)=Σ u∈V(G)d G(u)x ε′G (u), where ε′ G(u)=Σ v∈NG(u)εG(u)and d G(u) is the degree of u in G. In this paper modified eccentric connectivity polynomial is computed for several classes of composite graphs  

A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3-regular graphs are K4 and K3, 3. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r-regular graph, then G ϵ {Kr+1, Kr,r}. Also it is proved that for an even r, a connected triangle-free equimatchable r-regular graph is isomorphic to one of the graphs C5, C7, and Kr,r. © 2017 Wiley Periodicals, Inc  

The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree δ(G) whose adjacency matrix is non-singular, E(G)≥Δ(G)+δ(G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E(G). In this paper, first we show that E(L(G))≥|E(G)|+Δ(G)−5, where L(G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7. © 2021 Elsevier Inc  

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G)≥n. Here, we improve this result by showing that if G is a connected subcubic graph of order n≥8, then E(G)≥1.01n. Also, we prove that if G is a traceable subcubic graph of order n≥8, then E(G)>1.1n. Let G be a connected cubic graph of order n≥8, it is shown that E(G)>n+2. It was proved that if G is a connected cubic graph of order n, then E(G)≤1.65n. Also, in this paper we would like to present the best lower bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for... 

A Deza graph with parameters (n, k, b, a) is a k-regular graph with n vertices such that any two of its vertices have b or a common neighbours, where b ≥ a. In this paper we investigate spectra of Deza graphs. In particular, using the eigenvalues of a Deza graph we determine the eigenvalues of its children. Divisible design graphs are significant cases of Deza graphs. Sufficient conditions for Deza graphs to be divisible design graphs are given, a few families of divisible design graphs are presented and their properties are studied. Our special attention goes to the invertibility of the adjacency matrices of Deza graphs. © 2020, © 2020 Informa UK Limited, trading as Taylor & Francis Group  

A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children GA and GB of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in GA or GB if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. © 2021 Elsevier B.V  

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631{633) it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree Δ (G) whose adjacency matrix is non-singular, E(G) +δ (G) + Δ (G) and the equality holds if and only if G is a complete graph. Here, we prove the validity of this conjecture for planar graphs, triangle-free graphs and quadrangle-free graphs. © 2021 University of Kragujevac, Faculty of Science. All rights reserved  

A Deza graph G with parameters (n,k,b,a) is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children GA and GB of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in GA or GB if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. © 2021 Elsevier B.V  

The energy of a graph G, denoted by E(G), is defined as the sum of absolute values of all eigenvalues of G. In (MATCH Commun. Math. Comput. Chem. 83 (2020) 631{633) it was conjectured that for every graph G with maximum degree Δ(G) and minimum degree Δ (G) whose adjacency matrix is non-singular, E(G) +δ (G) + Δ (G) and the equality holds if and only if G is a complete graph. Here, we prove the validity of this conjecture for planar graphs, triangle-free graphs and quadrangle-free graphs. © 2021 University of Kragujevac, Faculty of Science. All rights reserved