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Research Article

Open Access Peer-reviewed

M. A. Rajan, Kinkar Ch. Das^{ }, V. Lokesha, I. Naci Cangül

Received April 17, 2019; Revised May 24, 2019; Accepted June 06, 2019

A is a labelled graph denoted by in which the vertex set of an has vertices labeled {} and edges such that there exist an edge between two distinct vertices labeled { and }, if { and } are coprime to each other. In this paper, some properties of the *ER*- are studied. An algorithm to compute *GCD *and* LCM *of any two numbers between and p by means of an *ER*- graph is also described.

**AMS Subject Classi****fi****cation 2000:** 11A41, 05C12, 05C07.

In number theory, the * totient* function or

This paper is organized into four sections. In section 2, main results are discussed. In section 3, algorithms to compute the greatest common divisor (GCD) and the least common multiplier (LCM) of any two given numbers between 1 and using *ER*- are described. Final section is for conclusions. For terminology and notations, one can look at ^{ 1, 5}. The following are the most used notations:

(1) is the set of prime numbers up to a real number *x*.

(2) is the GCD of two numbers *a* and *b* and is the LCM of them.

(3) A wheel graph is a graph with *p* vertices, formed by connecting a single vertex to all vertices of a cycle.

(4) Chromatic number of a graph *G* is the minimum number of colors to properly color the graph.

(5) Chromatic polynomial of a graph is the number of different ways of properly coloring a labeled graph by colors.

(6) Dominating set of a graph is a subset of such that every vertex of is adjacent to some vertex of .

(7) Domination Number is the cardinality of the smallest dominating set.

(8) is the line graph of .

(9) Permanent of a matrix is the sum , where runs over all permutations of the set

(10) Energy of a graph is the sum of the absolute eigenvalues of the adjacency matrix of that is, , where are the eigenvalues of ^{ 6}.

(11) Two graphs are said to be equi-energetic if and only if their energies are equal ^{ 7}.

(12) - is a variant of *ER* in which the vertices of the graph are labeled with the first prime numbers and .

In this section, we state some observations and results of an *ER*- graph.

(1) *ER* is a simple connected graph without pendant vertex.

(2) - .

(3) The trivial graph is an graph.

(4) and are the only complete graphs which are graphs.

(5) For , the *ER*- is not a regular graph.

(6) *ER*- has a star graph as its subgraph.

(7) For the* ER*-,

(i) The girth is 3.

(ii) The circumference is *p*.

(iii) The diameter is 2.

**Theorem 2.1.**** ***The number of edges in ER-** is*

*Proof*. By definition of an graph, there exists an edge between any two distinct vertices and of the *ER*- if and only if and . Thus, the vertex labelled with has number of edges with vertices in . Similarly the vertex labelled with has edges with vertices in {}, excluding the edge with vertex if any, since this edge is already counted within the edge set of vertex labeled by . Continuing this way for any arbitrary vertex labelled by , there are edges with vertices in excluding the edges with vertices from if any, as these edges are already accounted with the edge set of vertices of . Similarly the vertex labelled by 2 is having one edge with vertex labelled by 1 with excluding those edges which are associated with and finally vertex labeled by 1 has zero edges with excluding those edges which are associated with . Thus the total number of edges in ER- is .

**Corollary 2.2.**** **The number of edges in *ER*- is less than or equal to .

*Proof*. From Theorem 2.1,

and using the Euler's totient property, , we get

Thus

**Example 1.**** **The number of edges in *ER*- is The number of edges in *ER*- is , where is the number of edges in *ER*-.

**Theorem 2.3.** If and is odd, then *ER*- contains a spanning sub-graph as a wheel graph .

*Proof*. The proof can be given in two steps: 1) The *ER*- has cycles, and 2) Every vertex of has an edge with a distinguished vertex. Without loss of generality, let the vertex labelled by 1 be the distinguished vertex and the vertices be the possible vertices of . Since every vertex labelled by is coprime with the vertices labelled and . By the virtue of this, it has an edge with the vertices labeled by and and since is odd, the vertex labelled by 2 has an edge with vertex labelled by and 3. Thus these edges together with the vertices form a cycle . By definition, every natural number is co-prime to 1. Thus all the vertices labelled by of has an edge with the vertex labelled by 1. Hence *ER*- contains a wheel spanning subgraph .

**Corollary 2.4.** If and is even, then *ER*- contains a subgraph .

*Proof.** *Using Theorem 2.2, as is even, there can't be an edge between the vertices labelled by 2 and . Thus no cycle can be formed with the vertices . Thus an *ER*- contains a subgraph , whenever is even.

**Theorem 2.5.**** ***ER-** is also an ER-** graph and is isomorphic to ER-*.

*Proof.** **ER*- is a graph obtained by removing the vertex labelled by and all the edges associated with it. Thus the resultant graph has vertices and there exist an edge between any pair of vertices if and only if their labels are co-primes. Hence *ER*- is *ER*- graph having number of vertices and edges as and , respectively, and therefore, is isomorphic to *ER* graph.

**Theorem 2.6.**** ***If ** is the set of primes up to *,* then maximal clique in ER**-** is* .

*Proof.** *Since any two prime numbers are coprime, there exist edges between all the vertices labelled with prime numbers, and as these numbers are coprime with 1, they together form a complete graph with vertices. Thus the maximal clique in *ER*- is .

**Corollary 2.7.**** ***The chromatic number** ** of **ER**-** is **.*

*Proof.* By using Theorem 2.5, maximal clique in *ER**-* is . Hence it requires colors to color the prime labelled vertices and one more color to color the vertex labelled by . Since the remaining vertices can be colored by colors. Thus -) is .

**Theorem 2.8.**** ***Let ** be the set of prime numbers from 1 to ** with ** and ** **Then chromatic polynomial of ER-** is given by*

*Proof*. Let there are colors available to color graph *ER*- By corollary 2.7, for . Wlog, let and we color the vertices labled with , in ways respectively.

Let be the set of sets with the following properties.

Let ={vertex labeled with 1}. Let be the set of vertices which are labeled with the numbers which are divisible by prime . Then .

Let be the set of vertices which are labeled with the numbers which are divisible by prime and not divisible by prime . Then .

Wlog, for all in , let be the set of vertices which are labeled with the numbers which are divisible by prime and not divisible by primes , , , ... . Then

Further vertex of set can be colored in ways and for in , all the vertices of the set are not connected to each other and hence all the vertices of this set can be colred with the same color and thus every vertex of this set can be colored in ways. Thus in total all the vertices of can be clored in or ways.

**Theorem 2.9.**** **If pi is a prime number which is a label of a vertex v in ER- then the degree of v is

*Proof*. Let a vertex in ER- be labelled by a prime number , where . The maximum degree of any vertex in *ER* is . Then number of non-coprimes from 1 to with is . Since is a prime number and no number except 1 and itself can divide . Thus number of coprimes from 1 to with is . By the definition of *ER*, the degree of this vertex is

**Theorem 2.10.** The sum of degrees of vertices labelled with non prime vertices in is

(1) |

*where ** and each ** is a prime number*.

Proof. Let the vertices and are labelled with non-prime and prime numbers respectively. Also let be the set of numbers such that . It is well-known that the sum of degrees of all the vertices of any graph is twice the number of edges in it. By using this and Theorems 2.1 and 2.8,

i.e.,

i.e.,

This completes the proof.

**Theorem 2.11.**** **If then *ER*- always contains a Hamiltonian cycle.

*Proof*. *Case *1: Let be odd. By using mathematical Induction: it is true for the *ER* is , since is a Hamiltonian graph. Now assume that it is true for . We want to prove it for . By definition and Theorem 2.2, *ER* has a spanning subgraph as wheel graph . But a wheel graph has a Hamiltonian cycle. Thus *ER* has a Hamiltonian cycle.

*Case *2: Let be even and is the vertex set labeled with . By Corollary 2.3, ER contains a spanning subgraph then there exist a path and edge or path () between the vertices (labeled 1) and (labeled ). By joining these two paths, a cycle is formed, which proves the theorem for even also. Hence the result is true all values of . Note that (L(*ER*-) is also a Hamiltonian graph).

**Theorem 2.12.*** **The domination number and the total domination number of* *ER*- is --.

*Proof*. In *ER*-, the vertex labeled by 1 is adjacent to all the other vertices. Thus the set {1} is the minimum dominating set of $ER$- having the cardinality 1. Thus -. Similarly - .

**Theorem 2.13.**** ***If ** ER** is non-planar*.

*Proof*. By Kuratowaski's theorem on planarity, a graph is non-planar if and only if it contains either or as its subgraph, or obtains any of these subgraphs by means of homeomorphic operations. By Theorem 2.5, every *ER* has as its maximal clique. For , the number of primes is Thus *ER* has Kuratowaski's 2nd graph as its subgraph. Thus *ER* is non-planar.

**Remark 2.14.*** With **, no Kuratowaski's graph exists as a subgraph in ER** and also these graphs can be embedded in a plane** **surface. Hence they are planar graphs*.

**Theorem 2.15.**** ***If **, **, then ER-** has wheel** **subgraphs ** **,. Therefore, the number of** **wheel subgraphs is* .

*Proof.* By Theorem 2.4, it is true that, ER has *ER* as its subgraph and is also an graph. Since is odd, by Theorem 2.2, this graph has as its spanning subgraph. But the fact that is even implies *ER*- has no wheel subgraphs . But is odd and *ER* is also an graph and has as its subgraph. Thus alternating graphs have wheel subgraphs when the number of vertices is odd. Finally this process is continued till the number of vertices of reaches 5, which has as its subgraph. Thus *ER* has wheel subgraphs .

**Theorem 2.16.**** ***The graph **- ** is **.*

*Proof*. By the definition, vertices of this graph are labelled by prime numbers and one vertex labeled as 1. All these labels are coprime to each other and hence there exist an edge between every pair of vertices. Thus - is .

**Theorem 2.17.**** ***The number of edges in ER** is ** if ** **is prime, and ** if ** is composite, where ** is** **the number of edges in an ER-** graph*.

*Proof*. By construction method. Consider an - graph. Let be the vertex labeled by of the trivial graph . Now the graph *ER*- is constructed from *ER*- and by the following procedure. If is prime, then is coprime to all the numbers . *ER*- Then . If is composite, then is coprime to numbers among Then *ER*- is constructed by adding edges between the vertices of *ER*- and the vertex labeled of , such that the labels of the vertices of *ER*- are coprime with the vertex labeled of . Thus .

The adjacency matrix of - is defined by its entries if and 0 otherwise. Since is symmetric, its eigenvalues are real. Without loss of generality, we can write them as and call them the eigenvalues of .

**L****emma 3.1.** ^{ 8} *Let ** be a graph with sub-vertex** set ** having same set of neighbors **, where ** Then this** ** graph ** has at least ** equal eigenvalues *0*. Also the corresponding ** eigenvectors are*

**Theorem 3.2.*** **Let ER** be a graph of order** ** **. Also let ** be a prime number with positive integer ** ** such that **, **. Then** ** one of** **the eigenvalues of the adjacency matrix of an E**R**-** graph is ** of multiplicity **.*

*Proof.** *Since is an prime number, vertices with label have same set of neighbors. By Lemma 3.1, we conclude that ER- graph has at least equal eigenvalues 0, . This completes the proof.

**Theorem 3.3.** *If ** is a prime number, then one of the eigenvalues of the adjacency** **matrix of an ER**-** graph is **.*

*Proof.** *Since is a prime, there exist at-least two vertices labeled 1 and having degree . Then the adjacency matrix ( matrix) of the graph is given by

Now,

by applying the transform

by the determinant property.

To compute the eigenvalues of , write

From above, we get , that is, . Hence -1 is one of the eigenvalues of *ER*.

**Remark 3.4.*** **From the proof of Theorem **3.3**, we conclude that if there are ** vertices of ER** having degree ** then the multiplicity of eigenvalue -1 of ER** is *.

In mathematics, Bertrand's postulate (actually a theorem) was proven by several researchers (see, ^{ 9}). Using Bertrand's postulate, we prove the following result:

**Theorem 3.5.** *One of the eigenvalues of the adjacency matrix of ER**-** graph is* .

*Proof.** *If is prime, then the proof is same as Theorem 3.3. Otherwise, is composite. Then the proof is based on prime gaps. A prime gap is the difference between two successive prime numbers. To prove this theorem, the Bertrand's postulates ^{ 10, 11, 12} are applied. It states that there is always a prime number between and 2*x*. Any vertex in *ER*- has degree if it is labelled by 1 or a prime number such that = 1 for every such that is not a divisor of the numbers from to . By Bertrand's postulates, there is always a prime number between and as is composite. Without loss of generality, let be the nearest prime, which is less than and . Now is the least number divisible by . Since , then is not a divisor of numbers between to . Hence the degree of the vertex that is labelled by is . Thus there exist at least two vertices labeled by 1 and having degree . By applying Theorem 3.3 and Bertrand's postulates result is proven.

Let and be two graphs with and vertices, and and edges, respectively. The union of two graphs and , written , is the graph with vertex set and edge set . The join of graphs and with disjoint vertex sets and , and edge sets and is the graph union together with all the edges joining and . Thus, for example, \,, the complete bipartite graph.

**Theorem 3.6.**** ***One of the eigenvalues of the adjacency matrix of ** graph** *is zero and its multiplicity is at least for .

*Proof*. Let - By Theorems 3.3 and 3.5, there exist at least two vertices having degree in an , where is the number of vertices and is the number of edges in . Then we have

where is a graph of order with edges. Thus we have

Let be the characteristic polynomial of graph . From above one can see easily that

This completes the theorem.

The energy of the graph *G* is defined as

This quantity has a long known chemical application; for details see the surveys ^{ 13, 14, 15}. Recently much work on graph energy appeared also in the mathematical literature (see, for instance, ^{ 16, 17, 18, 19}).

**Theorem 3.7.** If is prime, then are equi-energetic graphs.

*Proof.* Denote by and .

Let , and , be the energies and adjacency matrices of two graphs and , respectively. Since is a prime and is co-prime with all numbers , this implies that in , the vertex labeled by has edges with all the vertices of . Thus we have

Let the eigenvalues of be . From above one can see easily that the eigenvalues of are and 0. Therefore . Hence the result follows.

**Theorem 3.8.**** ***Permanent of adjacency matrix of ** is zero*.

Let be the adjacency matrix of . Since the vertex labeled by 1 is adjacent to allthe vertices of the graph ER- and , it is not adjacent to none of the vertices of. By Theorems 3.3 and 3.5, this implies that has at least two rows with allzeros. Without loss of generality, let the 1st and th rows bezero rows. Using the definition of the permanent of matrix we get = = , where is a permutation over . In each of the products of sums, there exists at least one term which has elements of the and rows. Thus all the products of sums vanishes to zero. Hence the permanent of adjacency matrix is zero.

**Theorem 3.9.**** ***The number of edges of a line graph of ** is*

where is labelled with non-primes.

*Proof*. The number of edges of a $line$ $graph$ of is given by , where is the degree of the vertex of . By Theorem 2.1, for an , . The vertices of the graph can be partitioned into 3 sets: vertices labeled with prime numbers, vertices labeled with composite numbers and vertex labeled with 1 and their cardinalities are and 1, respectively. By using Theorems 2.1 and 2.3, the theorem is proved.

Let be the diagonal matrix of vertex degrees of graph Then the Laplacian matrix of is , where is the adjacency matrix of Let denote the eigenvalues of . They are usually called the Laplacian eigenvalues of . Among all eigenvalues of the Laplacian of a graph, the most studied is the second smallest, called the algebraic connectivity of a graph. It is well known that a graph is connected if and only if . Besides the algebraic connectivity, is the invariant that interested the graph theorists. The following two results are obtained in ^{ 20} and ^{ 21}.

**Lemma 3.10**. ^{ 20}* **Let ** be a graph with Laplacian spectrum **.** ** Then the Laplacian spectrum of ** is *}*, where** ** ** is the complement of the graph *.

**Lemma 3.11****.*** *^{ 21}* **Let ** be a graph with vertex **subset ** having the same set of neighbors** **, where** ** **. Then this graph ** ** has at least ** equal eigenvalues and they are all equal to the cardinality of the neighbor set. Also the corresponding ** ** eigenvectors are*

**Theorem 3.12**. *Let ER**-** be a graph of order **. Also let ** be a prime number** ** with positive integer ** ** such that **, **. Then the eigenvalues of the Laplacian matrix of an** **ER**-** graph is ** of **multiplicity ** ** Moreover, the largest two Laplacian eigenvalues are *.

*Proof.** *Since is an prime number, vertices with label have same set of neighbors. By Lemma 3.11 and Theorem 2.9, we conclude that *ER**-** ** *graph has at least equal eigenvalues ,

We have that two adjacent vertices are adjacent to all the remaining vertices of the graph *ER**-* by the proofs of Theorems 3.3 and 3.5. This implies that graph *ER**-** *has at least two vertices of degree , that is, graph has two isolated vertices. By Lemma 3.10, we conclude that the largest two Laplacian eigenvalues are . This completes the proof.

Though the is constructed using GCD concept, from its adjacency matrix, the GCD and LCM of any two numbers from 1 to can be computed with less complexity and this can easily be extended to higher order of the graphs. This is one of the techniques for finding GCD and LCM. The algorithm to compute the GCD of two numbers is as follows:

(1) Determine the adjacency matrix A of

(2) If A(a, b) = 1 or a = b, then (a,b) = 1.

(3) If then (a, b) = a or (if , then (a, b) = b); else find the common non neighbors of i and j, excluding 1.

(4) MinCommonNeighbour = minimum(common non-neighbors of i and j.)

(5) (a; b) = minimum(a, b; minCommonNeighbour)

(1) Determine the adjacency matrix *A* of

(2) If

(3) If or , then .

(4) Compute (*a*, *b*) using algorithm 3.1.

(5)

In this paper, we studied the spectral properties of co-prime labeled graphs () based on Euler's function. Several properties including connectivity, coloring and domination of these graphs are stated and proved. As part of further research, the study of generic spectral properties of these graphs is in progress.

K. C. Das was partially supported by the Faculty research Fund, Sungkyunkwan University, 2012 and Sungkyunkwan University BK21 Project, BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea.

[1] | D. M. Burton, Elementary Number Theory, 6th Edition, McGraw Hill, India (2007). | ||

In article | |||

[2] | Paul Erdös, Gabor. N.Sarkozy, On Cycles in the Coprime graph of integers, Electronic Journal of Combinatorics 4 (2) (1997) #R8. | ||

In article | |||

[3] | J. Baskar Babujee, Euler's Phi Function and Graph Labeling, Int. J. Math. Sciences 5 (20) (2010) 977-984. | ||

In article | |||

[4] | J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics 16 (2009) #DS6. | ||

In article | |||

[5] | F. Harary, Graph Theory, Addison-Wesley, Reading Mass, (1969). | ||

In article | View Article | ||

[6] | I. Gutman, The energy of a graph: old and new results, Algebraic Combinatorics and Applications, Springer Berlin, (2001), 196-211. | ||

In article | View Article | ||

[7] | K. C. Das and P. Kumar, Some new bounds on the spectral radius of graphs, Discrete Math. 281 (2004) 149-161. | ||

In article | View Article | ||

[8] | H. S. Ramane, H . B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Equienergetic graphs, Kragujevac, J. Math. 26 (2004), preceding paper. | ||

In article | |||

[9] | S. Ramanujan, A proof of Bertrand's postulate, Journal of the Indian Mathematical Society 11 (1919) 181-182. | ||

In article | |||

[10] | G.H. Hardy, An Introduction to the Theory of Numbers, 5th Edition, Oxford Science Publications, (1980). | ||

In article | |||

[11] | M. N. Huxley, On the Difference Between Consecutive Primes, Inventiones math. 15 (1972) 164-170. | ||

In article | View Article | ||

[12] | Thomas. R. Nicely, New Maximal Prime Gaps and First Occurences, Mathematics of Computation, AMS 68 (227) (1999) 1311-1315. | ||

In article | View Article | ||

[13] | I. Gutman, Total π-electron energy of benzenoid hydrocarbons, Topics Curr. Chem. 162 (1992) 29-63. | ||

In article | View Article | ||

[14] | I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer-Verlag, Berlin, 2001, pp. 196-211. | ||

In article | |||

[15] | I. Gutman, Topology and stability of conjugated hydrocarbons. The dependence of total π-electron energy on molecular topology, J. Serb. Chem. Soc. 70 (2005) 441-456. | ||

In article | View Article | ||

[16] | R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295. | ||

In article | View Article | ||

[17] | D. Stevanović, Energy and NEPS of graphs, Linear Multilinear Algebra 53 (2005) 67-74. | ||

In article | View Article | ||

[18] | D. Stevanović, I. Stanković, Remarks on hyperenergetic circulant graphs, Linear Algebra Appl. 400 (2005) 345-348. | ||

In article | View Article | ||

[19] | W. Yan, L. Ye, On the minimal energy of trees with a given diameter, Appl. Math. Lett. 18 (2005) 1046-1052. | ||

In article | View Article | ||

[20] | R. Merris, Laplacian matrices of graphs: A survey, Linear Algebra Appl. 197,198 (1994) 143-176. | ||

In article | View Article | ||

[21] | K. C. Das, Sharp Lower Bounds on the Laplacian eigenvalues of Trees, Linear Algebra Appl. 384 (2004) 155-169. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2019 M. A. Rajan, Kinkar Ch. Das, V. Lokesha and I. Naci Cangül

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

M. A. Rajan, Kinkar Ch. Das, V. Lokesha, I. Naci Cangül. On Some Properties of Coprime Labelled Graphs. *Turkish Journal of Analysis and Number Theory*. Vol. 7, No. 3, 2019, pp 77-84. http://pubs.sciepub.com/tjant/7/3/4

Rajan, M. A., et al. "On Some Properties of Coprime Labelled Graphs." *Turkish Journal of Analysis and Number Theory* 7.3 (2019): 77-84.

Rajan, M. A. , Das, K. C. , Lokesha, V. , & Cangül, I. N. (2019). On Some Properties of Coprime Labelled Graphs. *Turkish Journal of Analysis and Number Theory*, *7*(3), 77-84.

Rajan, M. A., Kinkar Ch. Das, V. Lokesha, and I. Naci Cangül. "On Some Properties of Coprime Labelled Graphs." *Turkish Journal of Analysis and Number Theory* 7, no. 3 (2019): 77-84.

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[1] | D. M. Burton, Elementary Number Theory, 6th Edition, McGraw Hill, India (2007). | ||

In article | |||

[2] | Paul Erdös, Gabor. N.Sarkozy, On Cycles in the Coprime graph of integers, Electronic Journal of Combinatorics 4 (2) (1997) #R8. | ||

In article | |||

[3] | J. Baskar Babujee, Euler's Phi Function and Graph Labeling, Int. J. Math. Sciences 5 (20) (2010) 977-984. | ||

In article | |||

[4] | J. A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics 16 (2009) #DS6. | ||

In article | |||

[5] | F. Harary, Graph Theory, Addison-Wesley, Reading Mass, (1969). | ||

In article | View Article | ||

[6] | I. Gutman, The energy of a graph: old and new results, Algebraic Combinatorics and Applications, Springer Berlin, (2001), 196-211. | ||

In article | View Article | ||

[7] | K. C. Das and P. Kumar, Some new bounds on the spectral radius of graphs, Discrete Math. 281 (2004) 149-161. | ||

In article | View Article | ||

[8] | H. S. Ramane, H . B. Walikar, S. B. Rao, B. D. Acharya, P. R. Hampiholi, S. R. Jog, I. Gutman, Equienergetic graphs, Kragujevac, J. Math. 26 (2004), preceding paper. | ||

In article | |||

[9] | S. Ramanujan, A proof of Bertrand's postulate, Journal of the Indian Mathematical Society 11 (1919) 181-182. | ||

In article | |||

[10] | G.H. Hardy, An Introduction to the Theory of Numbers, 5th Edition, Oxford Science Publications, (1980). | ||

In article | |||

[11] | M. N. Huxley, On the Difference Between Consecutive Primes, Inventiones math. 15 (1972) 164-170. | ||

In article | View Article | ||

[12] | Thomas. R. Nicely, New Maximal Prime Gaps and First Occurences, Mathematics of Computation, AMS 68 (227) (1999) 1311-1315. | ||

In article | View Article | ||

[13] | I. Gutman, Total π-electron energy of benzenoid hydrocarbons, Topics Curr. Chem. 162 (1992) 29-63. | ||

In article | View Article | ||

[14] | I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer-Verlag, Berlin, 2001, pp. 196-211. | ||

In article | |||

[15] | I. Gutman, Topology and stability of conjugated hydrocarbons. The dependence of total π-electron energy on molecular topology, J. Serb. Chem. Soc. 70 (2005) 441-456. | ||

In article | View Article | ||

[16] | R. Balakrishnan, The energy of a graph, Linear Algebra Appl. 387 (2004) 287-295. | ||

In article | View Article | ||

[17] | D. Stevanović, Energy and NEPS of graphs, Linear Multilinear Algebra 53 (2005) 67-74. | ||

In article | View Article | ||

[18] | D. Stevanović, I. Stanković, Remarks on hyperenergetic circulant graphs, Linear Algebra Appl. 400 (2005) 345-348. | ||

In article | View Article | ||

[19] | W. Yan, L. Ye, On the minimal energy of trees with a given diameter, Appl. Math. Lett. 18 (2005) 1046-1052. | ||

In article | View Article | ||

[20] | R. Merris, Laplacian matrices of graphs: A survey, Linear Algebra Appl. 197,198 (1994) 143-176. | ||

In article | View Article | ||

[21] | K. C. Das, Sharp Lower Bounds on the Laplacian eigenvalues of Trees, Linear Algebra Appl. 384 (2004) 155-169. | ||

In article | View Article | ||