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Introduction Let G be a simple connected graph with vertex set V(G) = {V1, V2,..., Vn}. The topological distance between vertices vi and vj of G is the length of a shortest path joining vi and vj. Let T be tree with k pendant (= vertices of degree one), labelled by V1, V2,..., Vk. The terminal distance matrix of T, is the square matrix of order k whose (i, j) -entry is the topological distance between vi and vj. The class of Kragujevac trees emerged in several studies addressed to solve the problem of characterizing the molecular graphs with respect to some topological indices such as the atom-bond connectivity index. Definition 1. Let P3 be the 3-vertex tree, rooted at one of its terminal vertices. For k = 2,3,..., construct the rooted tree Bk by identifying the roots of k copies of P3. The vertex obtained by identifying the roots of P3- trees is the root of Bk. Definition 2. Let d≥ 2 be an integer. Let β1, β2,..., βd be rooted trees specified in Definition 1. A Kragujevac tree T is a tree possessing a vertex of degree d, adjacent to the roots ofβ1, β2,..., βd. This vertex is said to be the central vertex of T, whereas d is the degree of T. The subgraphs β1, β2,..., βd are the branches of T. If all branches of T are isomorphic, T is called regular Kragujevac tree. Results and discussion Let ФG(λ) and PG(λ) denote the characteristic polynomial of the adjacency matrix and the Laplacian matrix of G. The following results are obtained in this paper. Theorem 1. Let k, d≥ 2 and Tk,d be a regular Kragujevac tree of order k and degree d. Then Φτκ,d (λ) = (λ2 - 1)(k-1)d[λ(λ² - k-1)]d-1(λ4 (k + d + 1)λ² + d), - Pτκ,d (λ) = λ(λ² - 3λ + 1)d(k-1)(λ³- (k + 4)λ² + (2k + 4)λ-1)d-1 (λ³ - (k + d + 4)λ² + (kd + 2k + 3d + 4)λ - 2kd - d - 1). Corollary 1. Let T be a Kragujevac tree of degree d, p denote the spectral radius and A denote the maximum degree of (non-central) vertices of T, then P ≤ √ Δ+d+1+√(Δ+ d + 1)²-4d/2 Corollary 2. Let T be a Kragujevac tree of degree d and E(T) denote the energy of T, then E(T) ≤ √2Δ+2d+2+2√(Δ + d + 1)² - 4Δ+2Δ+2d+2+2 √(Δ + d + 1)²- 4Δ+ 2d(A-1. +2d(Δ-1)+2(d-1) √Δ-1. Corollary 3. Let k, d≥ 2 and Tk,d be a regular Kragujevac tree of order k and degree d. The Kirshhof index of Tk,d is given as follows: Kf(Tk,d) =-d²(10k²+7k+1)-4d(k² + k). Conclusion ● The characteristic polynomial of adjacency and Laplacian matrix of the regular Kragujevac tree of order ݇ and degree ݀ is computed. ● An upper bound for the spectral radius and the energy of the Kragujevac trees is obtained. ● The Kirshhof index of Kragujejac tree of order ݇ and degree ݀ is computed. [ABSTRACT FROM AUTHOR]

Purpose: Cross-docking is a strategy in logistics management in which products from one or more suppliers are distributed directly to customers with a minimum loading or storage time. Cross docks act more as an inventory coordinator than as a storage warehouse. Finding a proper location is a critical issue in cross-docking. The page rank algorithm in social networks is one of the methods used to locate the accessible nodes in urban-networks. However, in the application of page rank algorithm in the location problems, only location access to the other nodes of the network has been considered, whereas, in cross docks’ location problem, the distance between the nodes, directly and indirectly, is of high importance due to the one-way nature of some routes and the average closeness of the cross docks to all nodes of the network in general. Thus, this research aims to develop the page rank algorithm in cross docks’ location problem, recognizing the distance between nodes with a weighted distanced adjacency matrix, besides considering the access and closeness to the different nodes of the network. Design/methodology/approach: The page rank algorithm developed by Agryzkov et al. (2012) in a way that the binary adjacency matrix of the path between nodes is changed into a weighted adjacency matrix, in which each element of the matrix in a row ‘i’ and column ‘j’ denotes the distance between ith and jth locations in the map. Besides, the main criteria of the Social Network Analysis (SNA) approach, such as Closeness centrality and Eigenvector centrality, also used to develop the model. Finally, the maximum covering location model in the network has been created by considering the output of the developed page rank algorithm as the input. To investigate the validity of the developed algorithm, it is applied in a practical cross docking problem and is solved by using the GAMS software. Findings: The developed page rank algorithm was applied to find the best locations of cross docks for an online store in the western half of district 3 in Tehran. The location points identified by the algorithm seem reasonable since they have access to the main highways allowing for easy and timely delivery of orders to customers distributed all over the investigated area. Research limitations/implications: Authors did not consider limitations such as the cost of renting and purchasing warehouse and the physical and traffic feasibility of the location points in the developed model. As a suggestion for future study, the model can be further developed by considering such features. Practical implications: By using the developed algorithm, the time of delivering goods and services to the customers by online stores would reduce, which not only increases the customers’ satisfaction but it also declines the company’s costs, significantly. Social implications: Timely delivery of orders to customers elevates customers’ feeling of safety and trust. Thus, it augments the online shopping habit of customers, leading to saving time, energy, and cost. Originality/value: Previously, the page ranking algorithm used in finding the best locations of advertising billboards in the urban map. Regarding the nature of a billboard location problem, only access/lack of access to a point determines the importance of a place, while in cross-docking location problem, in addition to access points, the distance of these warehouses, directly and indirectly, and the average proximity of them to all points of the network is of great importance. The algorithm developed in this study, in comparison with the previous version, not only considers the access of location points but also reflects the proximity of points through direct or indirect paths. [ABSTRACT FROM AUTHOR]

In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and Km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by Zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1,…,vn}. The distance between vertices vi and vj, denoted by d(vi,vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n×n matrix whose (i,j)-entry is d(vi,vj). The distance characteristic polynomial of Γ, denoted by χD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D-eigenvalues) of Γ. If λ is a D-eigenvalue of Γ with multiplicity m, then we denote it by λ[m]. Let λ1≥λ2≥…≥λn are the D-eigenvalues of Γ. Then λ1 is called distance spectral radius of Γ and we denote it by ρ(Γ). Also the multiset {λ1,…,λn} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S=S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay(G,S), is a graph with vertex set G and edge set {{g,sg}|g∈G,s∈S}. Cay(G,S) is a simple |S|-regular graph. Let x,y∈G. Then for all g∈G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g,h)=d(1,hg-1) and d(g)=d(1) for all g,h∈G, where d(x)=y∈G‍d(x,y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, Foster-Greenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ=Cay(G,S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of G. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, n-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group G admits a connected cubic distance integral Cayley graph if and only if G is isomorphic to one of the groups Z4, Z6, Z4×Z2, Z6×Z2, or Z2×Z2×Z2. Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4, K3,3, P3, P4 and P6, where Pn is the n-prism. Conclusion The following conclusions were drawn from this research. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for n-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group.

In this paper, graphs are undirected and loop-free and groups are finite. By Cn, Kn and km,n we mean the cycle graph with n vertices, the complete graph with n vertices and the complete bipartite graph with parts size m and n, respectively. Also by zn and Sn, we mean the cyclic group of order n and the symmetric group on n symbols, respectively. Let Γ be a simple connected graph with vertex set {v1, …, vn}. The distance between vertices vi and vj, denoted by d(vi, vj), is the length of a shortest path between them. The distance matrix of Γ, denoted by DΓ, is an n x n matrix whose (i,j)-entry is d;(vi, vj). The distance characteristic polynomial of Γ, denoted by XD(Γ) is det(λI-D) and its zeros are the distance eigenvalues (in short D;-eigenvalues) of Γ. If λ is a D;-eigenvalue of Γ with multiplicity m, then we denote it by λ ; [m] . Let λ ;1 ≥ λ ;2 ≥ λ n are the D-eigenvalues of Γ. Then λ ;1 is called distance spectral radius of Γ and we denote it by p;(Γ). Also the multiset {λ ;1, …, λ ;n} is denoted by SpecD(Γ). The studying of eigenvalues of distance matrices of graphs goes back to 1971, a paper by Graham and Pollack and thereafter attracted much more attention [2]. There are several applications of distance matrix such as the design of communication networks, network follow algorithms, graph embedding theory and in chemistry, for more details see [2]. Let G be a group and S = S-1 be a subset of G not containing the identity element of G. The Cayley graph of G with respect of S, denoted by Cay;(G, S), is a graph with vertex set G and edge set {{g, sg}|g ∈ G, s ∈ S}. Cay;(G, S) is a simple |S|-regular graph. Let x, y ∈ G. Then for all g ∈ G, x and y are adjacent if and only if xg and yg are adjacent. This implies that d(g, h) = d(1, hg −1 ) and d(g) = d(1) for all g, h ∈ G, where d(x) = ∑y∈G d(x, y). In the literature, the adjacency eigenvalues of Cayley graphs have been more widely used than the distance eigenvalues. A graph Γ is called distance (adjacency) integral if all the eigenvalues of its distance (adjacency) matrix are integers. A graph is called circulant if it is a Cayley graph over a cyclic group. Circulant graphs of valency 2 are cycles. In 2001, the distance eigenvalues of cycles computed [6]. In 2010, the distance spectra of adjacency integral circulant graphs characterized and proved that these graphs are distance integral [9]. In 2011, Rentlen discussed the distance eigenvalues of Cayley graphs of Coxeter groups using the irreducible representations of underlying group [10]. He proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. Then, FosterGreenwood and Kriloff proved that the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provided a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups [5]. In this paper, we determine the characteristic polynomial of the distance matrix of arbitrary Cayley graphs in terms of the irreducible representations of underlying groups. Let Γ = C𝑎𝑦(G, S) be a Cayley graph over a finite group G. It is well-known that one can determine the (adjacency) eigenvalues Γ by the irreducible representations of G, see for example [3, Corollary 7]. In this paper, by a similar argument, we determine the distance eigenvalues of Γ in terms of the irreducible representations of 𝐺. Then, as an application of our result, we exactly determine the distance eigenvalues of some well-know Cayley graphs: cycles, 𝑛-prims, hexagonal torus network and cubic Cayley graphs over abelian groups. Results and discussion We construct an infinite family of distance integral Cayley graphs. Also we prove that a finite abelian group 𝐺 admits a connected cubic distance integral Cayley graph if and only if 𝐺 is isomorphic to one of the groups Z4, Z6, Z4 × Z2, Z6 × Z2, or Z2 × Z2 × Z2 . Furthermore, up to isomorphism, there are exactly 5 connected cubic distance integral Cayley graphs over Abelian groups which are K4 , K3,3, P3, P4 and P4, where Pn is the 𝑛-prism. Conclusion The following conclusions were drawn from this research. The characteristic polynomial of the distance matrix of Cayley graphs over a group G is determined by the irreducible representations of G. Exact formulas for 𝑛-prisms, hexagonal torus network and cubic Cayley graphs over Abelian groups are given. Infinite family of distance integral Cayley graphs are constructed. Cubic distance integral Cayley graphs over finite abelian groups are classified. By a similar argument, one can find all quartic distance integral Cayley graphs over finite Abelian groups. One can easily compute the distance eigenvalues of a Cayley graph using irreducible representations of the underlying group. [ABSTRACT FROM AUTHOR]

Objective: Data generation in biology and biotechnology has greatly increased in recent years due to the very rapid development of high-performance technologies. These data are obtained from studying biological molecules, such as metabolites, proteins, RNA, and DNA, to understand the role of these molecules in determining the structure, function, and dynamics of living systems. Functional genomics is a field of research that aims to characterize the function and interaction of all the major components (DNA, RNA, proteins, and metabolites, along with their modifications) that contribute to the set of observable characteristics of a cell or individual (i.e., phenotype). Furthermore, in a breeding program, genetic improvement can be maximized through accurate identification of superior animals that are selected as parents of the next generation, thereby achieving breeding goals. Artificial neural networks have been proposed to alleviate this limitation of traditional regression methods and can be used to handle nonlinear and complex data, even when the data is imprecise and noisy. Omics data can be too large and complex to handle through visual analysis or statistical correlations. This has encouraged the use of machine intelligence or artificial intelligence. The objectives of this study was to review the main applications of artificial intelligence methods in functional genomics, cancer, agriculture, domestic animals and its intertwined fields, i.e. epigenomics, transcriptomics, epitranscriptomics, proteomics, and metabolomics, discuss important aspects of data management, such as data integration, cleaning, noise removal, balancing and ratio of missing data, functional genomicssystem modeling, artificial intelligence and systems biology, addressing legal, ethical and economic issues related to the application of artificial intelligence methods in the field of genomics and presenting a view of possible scenarios in the future. Materials and methods: In this review, all researches conducted in the field of artificial intelligence application in functional genomics, cancer, agriculture, domestic animals, and its intertwined fields, i.e. epigenomics, transcriptomics, epitranscriptomics, proteomics, and metabolomics, were tried, focusing on the applications of recent years after Increase production of big data to be studied and used. Results: The studies showed that the application of artificial intelligence in all fields, including functional genomics, cancer, agriculture, domestic animals, and its intertwined fields, i.e., epigenomics, transcriptomics, epitranscriptomics, proteomics, and metabolomics, is increasing rapidly and has many benefits. Conclusions: Considering the vital applications that are often addressed by biology and especially functional genomics, it is better to deal with artificial intelligence tools that are able to help mechanistic understanding of biological processes. In other words, enabling systems biology is important to reap the benefits of AI results in genomics. Interpretability can certainly help AI to be more easily adopted in practical applications such as medicine. In our opinion, increasing the volume and diversity of reliable big data and integrating it with theoretical modeling will help increase human trust in AI-based predictions and decisions in the future. On the one hand, model-based approaches can provide knowledge-based constraints. On the other hand, the results of artificial intelligence can help to create the parameters of models of biological systems. Functional genomics, as well as all fields of medicine, biology, agriculture, animal science, and other sciences that involve both individual and collective rights, is a complex field of research. Those who want to use AI in this field will find it difficult to navigate, as the field is sensitive to legal, ethical, and spiritual aspects. Artificial intelligence opens many opportunities that we should not refuse for fear of not understanding all the steps. The path that artificial intelligence will follow is just beginning to unfold. It has many promises and many potential dangers ahead. This path will probably be long and irreversible. Artificial intelligence will change our lives and we need to change our minds as soon as possible to adapt, accept, and manage the resulting changes in the best possible way, to ensure that they will bring as many benefits as possible and will cause the least possible negative consequences for us. [ABSTRACT FROM AUTHOR]

In this paper, relying on the clustering of complex networks that can determine large scale features of ‎the network, we study 48 financial markets across the world. To this end, we develop a modularity ‎maximization method for directed and weighted networks. According to the linear correlation measure, ‎we construct the adjacency matrix, and by using the theory of random matrices, we divide the space of ‎eigenvalues of our matrix into two irrelevant and relevant fragments. By considering the temporal ‎window and its evolution over time series, our results demonstrate that in the vicinity of so-called ‎financial crisis clusters, which are often affected by geographical characteristics, are formed and from the ‎perspective of complex networks, they show more random behavior‎.‎‎

Objectives: Due to the increasing share of consumption and watching videos - especially movies and series - in the basket of Iranian households, several systems have been set up to facilitate people's access to these videos. One of the most important types of these systems is the video-on-demand system which has taken unprecedented growth in attracting audiences in recent years. Just as the multiplicity of content in these systems causes users to be diverse and satisfied, this multiplicity can be more confusing for them to find interesting content. Therefore, the need for recommendation systems to further predict user interests and provide consistent content is felt more and more day by day. The purpose of this study is to provide an efficient method of recommending videos based on user viewing data in the video-on-demand system. Methods: In this research, a new bidding algorithm based on users' tastes and video watching data in the video-on-demand system is presented. This algorithm is based on the concepts and indicators of social network analysis. How this algorithm works is that first the similarity of the videos is calculated based on the percentage of movies viewed by the user and based on that the similarity matrix of the movies is formed. In the next step, based on the similarity matrix of the films, the communication graph of the films is formed, and in the next step, while discovering the communities in the graph, the centrality indicators of each film are calculated. Then, based on the viewing data of each user and the history of his favorite and nonfavorite movies, the members of the community with the most favorite movies are selected as candidates for the user and after calculating their distance from the user's favorite and nonfavorite movies, and if the final index is positive, Bids are offered to the user. Resultds: The performance of the proposed algorithm of this research is evaluated on the data of 50, 100, and 200 users of the CinemaMarket site, the results of which show the better performance of the proposed algorithm of this research in comparison with the algorithms of Naive Bayes, k-nearest neighbors and ID3. Conclusions: Recommendation systems offer content by having a lot of information from users and their behavioral records. The proposed algorithm of this research can perform the task of suggesting the desired content of users with the least possible information, ie information related to users' observations with desirable and acceptable performance. [ABSTRACT FROM AUTHOR]

Robustness is considered as one of the important and vital features in today’s complex networks and advanced infrastructures and has become a popular and growing research field in the recent years. This research field seeks to find solutions and mechanisms to improve the connectivity of networks against random failures and systematic attacks. Until now, various different measures have been suggested for measurement and assessment of robustness and resilience of complex networks. An important part of these measures is dedicated to the algebraic graph theory and the study and examination of the spectrum of the graph’s adjacency matrix (graph spectrum). The energy of any simple graph is the sum of the absolute values of its eigenvalues. In this paper, it is first shown that the energy of any network has a strong correlation with its robustness, then; a rewiring mechanism for optimization of a graph’s energy measure is presented. Using an evolutionary algorithm (genetic), we try to find that rewiring which set of edges causes the most energy increase compared to the original graph. By answering this question, we can decide about finding an optimal solution to optimize the energy and connectivity of complex networks in different areas of complex and social networks, and; as a result, increase their robustness and resilience. The numerical results derived from simulations provide us with knowledge of the location and number of edges that should be rewired to gain a maximal increase in energy, and; as a result, a maximal increase in robustness.

Aim: The study purpose was visualization and analysis of word co-occurrence network of Journal of Applied Psychology of Shahid Beheshti University. Method: Research method was descriptive and using scientometric indices and social network analysis using a purposeful method. Research population consisted of all articles published in the mentioned Journal during period of 2015-16 (N= 287). Title, abstract and key words of 287 articles were reviewed and categorized under the guidance of two psychologists in various subject areas. In the next step, adjacency matrix of conjugate terms of extracted words was obtained in Excel software. Finally, data was generated using the U.S.I.Net software. She. S Vivor was analyzed. Results: Results showed that domains of cognitive psychology, social issues, emotion and personality have been of great importance in this Journal. Most articles in the Journal of Applied Psychology have been published in the field of cognitive psychology. In addition, the results showed that self-efficacy, coping styles, and adaptability had the highest degree of centrality degree and the components of subjective well-being, self-efficacy, adaptability and social dimensions were the most important indicator of the centrality of vision. Conclusion: The presentation of the thematic map of the articles in the journal of applied psychology will make the researchers more aware of the status of the researchers conducted and their relationship with the knowledge of the subject's gaps and help researchers to understand the general view of the scientific framework of the field of psychology and help researchers to select hot topics in this area