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7. Steiner (revised) Szeged index of graphs

Author

Ghorbani, Modjtaba, Li, Xueliang, Maimani, Hamid Reza, Mao, Yaping, Rahmani, Shaghayegh, and Rajabi-Parsa, Mina

Subjects
Mathematics - Combinatorics, 05C05, 05C12, 05C35, 92E10
Abstract

The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least 2 and $S\subseteq V(G)$, the Steiner distance $d_G(S)$ of the set $S$ of vertices in $G$ is the minimum size of a connected subgraph whose vertex set contains or connects $S$. In this paper, we introduce the concept of the Steiner (revised) Szeged index ($rSz_k(G)$) $Sz_k(G)$ of a graph $G$, which is a natural generalization of the well-known (revised) Szeged index of chemical use. We determine the $Sz_k(G)$ for trees in general. Then we give a formula for computing the Steiner Szeged index of a graph in terms of orbits of automorphism group action on the edge set of the graph. Finally, we give sharp upper and lower bounds of ($rSz_k(G)$) $Sz_k(G)$ of a connected graph $G$, and establish some of its properties. Formulas of ($rSz_k(G)$) $Sz_k(G)$ for small and large $k$ are also given in this paper., Comment: 12 pages

Published
2019

8. The general position problem on Kneser graphs and on some graph operations

Author

Ghorbani, Modjtaba, Klavžar, Sandi, Maimani, Hamid Reza, Momeni, Mostafa, Mahid, Farhad Rahimi, and Rus, Gregor

Subjects
Mathematics - Combinatorics
Abstract

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number (gp-number) ${\rm gp}(G)$ of $G$. The gp-number is determined for some families of Kneser graphs, in particular for $K(n,2)$ and $K(n,3)$. A sharp lower bound on the gp-number is proved for Cartesian products of graphs. The gp-number is also determined for joins of graphs, coronas over graphs, and line graphs of complete graphs.

Published
2019

15. The General Position Problem on Kneser Graphs and on Some Graph Operations

Author

Ghorbani Modjtaba, Maimani Hamid Reza, Momeni Mostafa, Mahid Farhad Rahimi, Klavžar Sandi, and Rus Gregor

Subjects
general position set, kneser graphs, cartesian product of graphs, corona over graphs, line graphs, 05c12, 05c69, 05c76, Mathematics, QA1-939
Abstract

A vertex subset S of a graph G is a general position set of G if no vertex of S lies on a geodesic between two other vertices of S. The cardinality of a largest general position set of G is the general position number (gp-number) gp(G) of G. The gp-number is determined for some families of Kneser graphs, in particular for K(n, 2), n ≥ 4, and K(n, 3), n ≥ 9. A sharp lower bound on the gp-number is proved for Cartesian products of graphs. The gp-number is also determined for joins of graphs, coronas over graphs, and line graphs of complete graphs.

Published
2021
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29. Nullity of Graphs

Author

Ghorbani, Modjtaba, Songhori, Mahin, Cataldo, Franco, Series editor, Milani, Paolo, Series editor, Ashrafi, Ali Reza, editor, and Diudea, Mircea V., editor

Published
2016
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31. On the Graovac-Pisanski index

Author

Hakimi-Nezhaad Mardjan and Ghorbani Modjtaba

Subjects
Wiener index, polyhedral graphs, Science
Abstract

The Graovac-Pisanski index (GP index) is an algebraic approach for generalizing the Wiener index. In this paper, we compute the difference between the Wiener and GP indices for an infinite family of polyhedral graphs.

Published
2017
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39. GRAPHS WITH AT MOST FOUR SEIDEL EIGENVALUES.

Author

GHORBANI, MODJTABA, HAKIMI-NEZHAAD, MARDJAN, and BO ZHOU

Subjects
EIGENVALUES, PROBLEM solving
Abstract

Let G be a graph of order n with adjacency matrix A(G). The eigenvalues of matrix S(G) = Jn - In - 2A(G), where Jn is the n by n matrix with all entries 1, are called the Seidel eigenvalues of G. Let G(n, r) be the set of all graphs of order n with a single Seidel eigenvalue with multiplicity r. In the present work, we will characterize all graphs in the class G(n, n - i) for i = 1, 2 and for the case i = 3 our characterization is done by this condition that the nullity of S(G) is zero. If the nullity of S(G) is not zero the problem is solved in special cases. [ABSTRACT FROM AUTHOR]

Published
2023
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44. On edge-path eigenvalues of graphs.

Author

Akbari, Saieed, Azizi, Seyran, Ghorbani, Modjtaba, and Li, Xueliang

Subjects
EIGENVALUES, MATRICES (Mathematics), CHARTS, diagrams, etc., REGULAR graphs
Abstract

Let G be a graph with vertex set V (G) = { v 1 , ... , v n } and E P (G) be an n × n matrix whose (i , j) -entry is the maximum number of internally edge-disjoint paths between v i and v j , if i ≠ j , and zero otherwise. Also, define E P ¯ (G) = E P (G) + D , where D is a diagonal matrix whose i-th diagonal element is the number of edge-disjoint cycles containing v i , whose E P (G) is a multiple of J−I. Among other results, we determine the spectrum and the energy of the matrix E P ¯ (G) for an arbitrary bicyclic graph G. [ABSTRACT FROM AUTHOR]

Published
2022
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45. Network Analyzing by the Aid of Orbit Polynomial

Author

Ghorbani, Modjtaba and Dehmer, Matthias

Subjects
QA1-939, polynomial roots, orbit-stabilizer theorem, group action, orbit, Mathematics
Abstract

This article aims to be a further contribution to the research on structural complexity networks. Here, we emphasize measures to determine symmetry. The so-called “orbit polynomial” is defined by OG(x)=∑iaixi, where ai is the number of orbits of size i. Furthermore, the graph polynomial 1−OG(x) has a unique positive root in the interval (0,1), which can be considered as a relevant measure of the symmetry of a graph. In the present paper, we studied some properties of the orbit polynomial with respect to the stabilizer elements of each vertex. Furthermore, we constructed graphs with a small number of orbits and characterized some classes of graphs in terms of calculating their orbit polynomials. We studied the symmetry structure of well-known real-world networks in terms of the orbit polynomial.

Published
2021

46. Orbit Polynomial of Graphs Versus Polynomial with Integer Coefficients

Author

Ghorbani, Modjtaba, Jalali-Rad, Maryam, and Dehmer, Matthias

Subjects
TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, QA1-939, ComputerApplications_COMPUTERSINOTHERSYSTEMS, polynomial roots, orbit-stabilizer theorem, Astrophysics::Earth and Planetary Astrophysics, group action, orbit, Mathematics
Abstract

Suppose ai indicates the number of orbits of size i in graph G. A new counting polynomial, namely an orbit polynomial, is defined as OG(x) = ∑i aixi. Its modified version is obtained by subtracting the orbit polynomial from 1. In the present paper, we studied the conditions under which an integer polynomial can arise as an orbit polynomial of a graph. Additionally, we surveyed graphs with a small number of orbits and characterized several classes of graphs with respect to their orbit polynomials.

Published
2021

48. Orbit Entropy and Symmetry Index Revisited

Author

Jalali-Rad, Maryam, Ghorbani, Modjtaba, Dehmer, Matthias, Emmert-Streib, Frank, Tampere University, and Computing Sciences

Subjects
QA1-939, symmetry index, automorphism group, entropy, 113 Computer and information sciences, Mathematics, MathematicsofComputing_DISCRETEMATHEMATICS
Abstract

The size of the orbits or similar vertices of a network provides important information regarding each individual component of the network. In this paper, we investigate the entropy or information content and the symmetry index for several classes of graphs and compare the values of this measure with that of the symmetry index of certain graphs. publishedVersion

Published
2021

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