Some bounds for total communicability of graphs

Accepted Manuscript Some bounds for total communicability of graphs

Kinkar Ch. Das, Mohammad Ali Hosseinzadeh, Samaneh Hossein-Zadeh, Ali Iranmanesh

PII: DOI: Reference:

S0024-3795(19)30047-3 https://doi.org/10.1016/j.laa.2019.01.023 LAA 14868

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Linear Algebra and its Applications

Received date: Accepted date:

2 June 2018 26 January 2019

Please cite this article in press as: K.Ch. Das et al., Some bounds for total communicability of graphs, Linear Algebra Appl. (2019), https://doi.org/10.1016/j.laa.2019.01.023

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Some Bounds for Total Communicability of Graphs Kinkar Ch. Dasa , Mohammad Ali Hosseinzadehb , Samaneh Hossein-Zadehb , Ali Iranmaneshb,∗ a

Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea.

b

Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran.

Abstract In a network or a graph, the total communicability (T C) has been defined as the sum of the entries in the exponential of the adjacency matrix of the network. This quantity offers a good measure of how easily information spreads across the network, and can be useful in the design of networks having certain desirable properties. In this paper, we obtain some bounds for total communicability of a graph G, T C(G), in terms of spectral radius of the adjacency matrix, number of vertices, number of edges, minimum degree and the maximum degree of G. Moreover, we find some upper bounds for T C(G) when G is the Cartesian product, tensor product or the strong product of two graphs. In addition, Nordhaus-Gaddum-type results for the total communicability of a graph G are established. Keywords: Matrix function, Total communicability, Spectral radius, Regular graph, Nordhaus-Gaddum-type results AMS Subject Classification Number: 05C50, 15A16, 05C12 ∗

Corresponding author

E-mail addresses: [email protected] (K. Ch. Das), [email protected] (M. A. Hosseinzadeh), [email protected] (S. Hossein-Zadeh), [email protected] (A. Iranmanesh)

1

1

Introduction

Let G = (V, E) be a simple connected graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G), where |V (G)| = n and |E(G)| = m are the order and the size of G, respectively. Let G be the complement of graph G. The adjacency matrix of a graph G with n vertices is a (0, 1)-matrix A(G) = [aij ]n×n , where aij = 1, if there is an edge between vi and vj , and aij = 0, otherwise. Let λ1 (G) ≥ λ2 (G) ≥ · · · ≥ λn (G) denote the eigenvalues of A(G). They are usually called the adjacency eigenvalues of graph G. The spectral radius of A(G) is denoted by λ(G) and is the largest eigenvalue of A(G). Also we denote the minimum eigenvalue of A(G) by λmin (G). For a real symmetric matrix B, the Perron vector of B is the unique unit positive vector x such that Bx = λ(B)x, where λ(B) is the spectral radius of B. Matrix functions have an important influence on the study of networks [12, 16]. For example, the concepts of communicability, subgraph centrality and Katz index are derived from matrix functions f (A) of the adjacency matrix and allow the characterization of local structural properties of networks (see [11, 13, 14, 15, 21]). The trace of f (A), which is known as the Estrada index of a graph with adjacency matrix A (see [25]) is a useful characterization of the global structure of a graph and it has found applications as an index of natural connectivity for studying robustness of networks [26, 27]. The greatest appeal of the use of functions of the adjacency matrix for studying graphs is that when representing them in terms of a Taylor function expansion:  k f (A) = ∞ k=0 ck A(G) , the entries of the k-th power of the adjacency matrix provides information about the number of walks of length k between the corresponding pair of not necessarily different nodes (see next section for formal definitions). Then, the important ingredient of the definition of f (A) lies in the use of the coefficients ck . The use of ck = k!−1 gives rise to the exponential function of the adjacency matrix, which is the basis of the communicability/subgraph centrality. The communicability function of G between two vertices vi and vj is defined by a matrix function as follows [10, 12, 13, 14]:   ∞ A(G)k n   ij k (vj ), k (vi )ψ Gij = = (exp(A(G)))ij = e λk ψ k! k=0

k=1

k is the eigenvector corresponding to eigenvalue λk of the adjacency matrix where ψ 2

A(G). The communicability function is an important quantity for studying communication processes in networks. It counts the total number of walks starting at node vi and ending at node vj , weighted in decreasing order of their length by a factor

1 k! ,

therefore it is considering shorter walks more influential than longer one. The Gii terms of the communicability function characterize the degree of participation of a node in all subgraphs of the network, giving more weight to the smaller ones. Thus, it is known as the subgraph centrality of the corresponding node [15]. The following quantity is known in the algebraic graph theory literature as the Estrada index of a graph with n vertices: EE(G) =



Gii = tr(exp(A(G))) =

n 

eλk ,

k=1

vi ∈V (G)

which is a characterization of the global properties of a network. In this paper, in Theorem 3, we show that a Rayleigh quotient like result holds for a matrix function of a real symmetric matrix. For details about Rayleigh quotient, we refer to [7, pp. 11]. Total communicability of a network or a graph such as G, defined as the sum of the entries in the exponential of the adjacency matrix of the network [2]. It provides a global measure of how well the nodes in a graph can exchange information. Communicability is based on the number and length of graph walks connecting pairs of nodes in the network. Pairs of nodes vi , vj with high communicability correspond to large entries [eA ]ij in the matrix exponential of A(G), the adjacency matrix of the network. Total network communicability can also be used to measure the connectivity of the network as a whole. For instance, given two alternative network designs with a similar budget in terms of number of candidate edges, one can compare the two designs by computing the respective total communicabilities and pick the network with the highest one, assuming that a well-connected network with high node communicability is the desired goal. In  the sequel, we focus on the quantity T C(G) = (vi , vj )∈V (G)×V (G) Gij and we find some bounds for T C(G) in terms of λ(G). In Theorem 4, we obtain an explicit formula for T C(G) in terms of eigenvalues and the eigenvectors of G. Also, for a connected graph G of order n and spectral radius λ(G), we give an upper bound on T C(G) and characterize the extremal graphs. Furthermore, we present a lower bound on T C(G) in terms of λ(G) and ω(G) (ω(G) is the clique number of G which we define it in the next section), and the extremal graph is also characterized. Moreover, we obtain a

3

Nordhaus-Gaddum-type results for total communicability of a graph. Finally, as a consequence of Theorem 4, we obtain some upper bounds for the total communicability of some important graph products, such as Cartesian product, tensor product and the strong product. Likewise, we compute some upper bounds for T C(G) in terms of the number of vertices, number of edges, maximum degree and the minimum degree of a graph G. In the last section we give some computations for variety of connected graphs, with the aim of comparing the bounds for total communicability.

2

Preliminaries

In this paper we follow the definitions and terminologies of [4, 7]. Let di be the degree of the vertex vi and Ni be the neighbor set of vertex vi (i = 1, 2, . . . , n). Therefore |Ni | = di (|X| is the cardinality of the set X). The minimum vertex degree is denoted by δ(G) and the maximum by Δ(G). One of the oldest and most studied topological indices in mathematical chemistry, is called the first Zagreb index [20], defined as follows: M1 = M1 (G) =

n 

d2i =

i=1



(di + dj ),

vi vj ∈E(G)

where di is the degree of the vertex vi in G. A walk W of length k from a vertex u to a vertex v is a sequence of vertices W : u = v0 , v1 , . . . , vk = v, where vi−1 vi is an edge of G for i = 1, . . . , k. The number of walks of length k − 1, i.e., number of walks with k vertices, is denoted by wk . Denote by Kn , Pn , Cn , the complete graph, path and cycle on n vertices, respectively. Also Snk , (n ≥ k ≥ 3), is the graph obtained from a cycle Ck by attaching (n − k) pendent edges to only one vertex of the cycle. By Kp,q we denote the complete bipartite graph with two partitions of orders p and q (p + q = n). A clique of a graph G is a complete subgraph of G. The maximum order of a clique of G is called the clique number of G and is denoted by ω(G). A tree is a connected acyclic graph. Also, a graph with exactly one cycle is called unicyclic. It is known that the number of walks of length t in a graph G between two vertices vi and vj is the ij-th entry of the matrix At , i. e., (At )ij . The all 1 vector is denoted by j. Lemma 1. [6] Let G be a simple graph on n vertices. Then √ 1+ 3 n < 1.37n. λ(G) + λ(G) ≤ 2 4

The following result has been obtained in [8]. Lemma 2. [8] Let G be a graph and V (G) = {v1 , v2 , . . . , vn }. Then n 

|Ni ∩ Nj | =

j=1 j=i



(dj − 1) = di mi − di , vi ∈ V (G),

vj :vi vj ∈E(G)

where mi is the average degree of the vertices adjacent to vi and |Ni ∩ Nj | is the number of common neighbors of vi and vj .

3

New Results

In this section, we bring our main results. We start with a theorem which states a generalization of Rayleigh quotient for a real symmetric matrix and we expect to see it for a matrix function. Theorem 3. Let B be a real symmetric matrix and f be a real function such that it  f (k) (α) k has a Taylor series expansion f (z) = ∞ k=0 k! (z − α) with radius of convergence

r. If |λ(B) − α| < r, then for each real vector x we have xT f (B)x ≤ f (λ(B)) x2

and equality holds if and only if x is a constant multiple of the Perron vector of B. Furthermore, if |λmin (B) − α| < r, then xT f (B)x ≥ f (λmin (B)) x2 for each real vector x and in this case, the equality holds if and only if x is a λmin (B)eigenvector of B. Proof. Consider the matrix series f (B) =

∞

k=0

f (k) (α) k k! (B − αI) .

Since |λ(B) − α| < r,

the matrix series f (B) is convergent (see Theorem 4.7 of [16]). As a consequence of Rayleigh quotient [7, pp. 11], for each positive integer k and a real vector x we have xT (B − αI)k x ≤ (λ(B) − α)k x2

5

and equality holds if and only if x is an eigenvector of B−αI corresponding to eigenvalue λ(B) − α. Thus

xT f (B)x x2

=

=

xT



 − αI)k x

∞  f (k) (α) xT (B − αI)k x k=0

≤

∞ f (k) (α) k=0 k! (B x2

k!

∞  f (k) (α) k=0

k!

x2 (λ(B) − α)k

= f (λ(B)). Notice that in the last equation, the condition |λ(B) − α| < r ensures convergence of the series to f (λ(B)). Now, if the equality holds, then for each positive integer k we have

xT (B−αI)k x x2

= (λ(B) − α)k , and this occurs if and only if x is a constant multiple

of the Perron vector of A. Now, suppose that |λmin (B) − α| < r. The matrix series f (B) is convergent and by Rayleigh quotient, for each positive integer k and real vector x, we have (λmin

(B) − α)k .

xT (B−αI)k x x2

≥

xT f (B)x x2

≥

Also the equality holds if and only if x is an (λmin (B) − α)-eigenvector

of B−αI. Since |λmin (B)−α| < r, similar to previous discussion we find that

f (λmin (B)) and the equality holds if and only if x is a λmin (B)-eigenvector of B, as desired. In [2, Proposition 1], the following bounds for total communicability of a connected graph G with order n has been obtained. EE(G) ≤ T C(G) ≤ neλ(G) Now, in the next theorem we find an explicit formula for total communicability of G and we give another lower bound for T C(G). Also, we determine the graphs G of order n such that their total communicabilities are equal to neλ(G) . Theorem 4. Suppose that G is a connected graph of order n. 6

(a) Let λ1 , . . . , λn be the eigenvalues of G with the corresponding orthogonal unit  eigenvectors x1 , . . . , xn . Also let xi = (xi1 , . . . , xin ), set ci = ( nj=1 xij )2 . Then T C(G) =

n 

c i eλi .

i=1

(b) Then T C(G) ≤ neλ(G) and the equality holds if and only if G is a regular graph.   ω(G) λ(G)eλ(G) . Moreover, if the equality holds, then G is a (c) Then T C(G) ≥ ω(G)−1 regular graph or a regular complete ω(G)-partite graph. Proof. Let x = (x1 , . . . , xn )T be a real column vector and let B = [bij ] be a real matrix of order n. One can find that 

xT Bx =

bpq xp xq .

1≤p,q≤n

Thus we have







jT A(G)k j =

A(G)k



vp ∈V (G) vq ∈V (G)

pq

.

Now, we claim that T C(G) = jT eA(G) j. To see this, for the graph G with adjacency matrix A(G) we can write 

T C(G) =

Gpq =

  ∞ A(G)k   pq



vp ∈V (G) vq ∈V (G) k=0

(vp ,vq )∈V (G)×V (G)

=

∞ T  j A(G)k j k=0

k!

k!

= jT eA(G) j.

(1)

If x1 , . . ., xn is a complete system of mutually orthogonal normalized eigenvectors of A(G) belonging to the spectrum [λ1 , . . . , λn ], X = (x1 , . . . , xn ), and D is the diagonal matrix diag(λ1 , . . . , λn ), then X is orthogonal (i.e., X −1 = X T ) and A(G) = XDX T . Now consider the matrix function eA(G) . One can see that the eigenvalues of eA(G) are eλ1 , . . . , eλn with corresponding eigenvectors x1 , . . . , xn . Thus, we get the following: T A(G)

T C(G) = j e

j=j

T





XD X

T



T

T

j = [j x1 , . . . , j xn ]D



[xT1 j, . . . , xTn j]T

=

n  i=1

7

ci eλi ,

where D is the diagonal matrix diag(eλ1 , . . . , eλn ) and the proof of (a) is completed. To prove (b), by Theorem 3, we have T C(G) = jT eA(G) j ≤ eλ(G) j2 = n eλ(G) . Also the equality holds if and only if j is an eigenvector of G and this one occurs if and only if G is regular (see [7, Proposition 1.1.2]), as desired. Now, we prove (c). In [23] it is shown that for a graph G with n vertices and a positive integer k ≥ 1, λk (G) ≤

ω(G)−1 ω(G) wk .

Moreover, the following result has been proved in

the same paper: Suppose that G is a graph such that λk (G) =

ω(G)−1 ω(G) wk

(k ≥ 1). If k = 1, then G

is regular complete ω(G)-partite. If k > 1, then G has a single non-trivial component H. If ω(G) > 2, then H is a regular complete ω(G)-partite graph. If ω(G) = 2, then H is complete bipartite, and if k is odd, then H is a regular graph. Using these results we have T C(G) =



Gpq =

∞  wk+1 k=0

(p,q)∈V (G)×V (G)

k!

 ∞  ω(G) 1 λk+1 (G) ≥ k! ω(G) − 1 k=0

 =  =

ω(G) ω(G) − 1 ω(G) ω(G) − 1

λ(G)

∞  λk (G) k=0



k!

λ(G)eλ(G) . 

Also if the equality holds, then for every non-negative integer k, ωk+1 =

ω(G) ω(G)−1



λk+1 (G)

and so we are done. Remark 5. It is known that the spectral radius of an r-regular graph is r. Now, let G be a graph with n vertices such that λ(G) ≤ r. As a consequence of Theorem 4, we have T C(G) = ner if and only if G is r-regular. 8

For a graph G, the chromatic number χ(G) is the minimum number of colors needed to color the vertices of G in such a way that no two adjacent vertices are assigned the same color. In 1956, Nordhaus and Gaddum [24] gave bounds involving the chromatic number χ(G) of a graph G and its complement G . They proved that √ 2 n ≤ χ(G) + χ(G) ≤ n + 1

and

n ≤ χ(G) · χ(G) ≤

(n + 1)2 . 4

Furthermore, these bounds are best possible for infinitely many values of n. Initially, this type of relation did not attract much attention. Actually, the first detailed study of these relations came almost a decade after the publication of the paper of Nordhaus and Gaddum [24]. Since then many results of this kind were obtained, for a survey see [1]. In the next theorem, a Nordhaus-Gaddum-type results for total communicability of a connected graph is given. Theorem 6. Let G be a connected graph of order n with m edges and connected complement G. Then   1 n(n2 + 3) + M1 (G) − 2(n − 1)m < T C(G) + T C(G) < n e + e1.37n−1 2

(2)

and     M (G) + n(n2 + 3) 1 1 − m(n − 1) M1 (G) + n2 (n2 + 1) + mn (n − 1)2 − 4m 4 2 < T C(G) T C(G) < n2 e1.37n ,

(3)

where M1 (G) is the first Zagreb index of the graph G. Proof. Let us consider a function f (x) = ex + e1.37n−x

for 1 ≤ x ≤ n − 1.

Then f  (x) = ex − e1.37n−x . Therefore f (x) is an increasing function on [1.37n/2, n − 1] and is a decreasing function on [1, 1.37n/2]. Hence x

e +e

1.37n−x



1.37n−1

= f (x) ≤ max e + e

9

,e

n−1

+e

0.37n+1



= e + e1.37n−1 .

By applying this inequality, Theorem 4 (b) and Lemma 1, we have       T C(G) + T C(G) ≤ n eλ(G) + eλ(G) < n eλ(G) + e1.37n−λ(G) ≤ n e + e1.37n−1 , which gives the required result in the right inequality of (2). The ik-th entry of A(G)2 is

if i = k,

di

|Ni ∩ Nk | Otherwise. By Lemma 2 with

n  i=1

d2i =

T

n  i=1

di mi , one can see that

2

j A(G) j =

n 

di +

i=1

n n  

|Ni ∩ Nk |

i=1 k=1, k=i

= 2m +

n 

(di mi − di )

i=1

=

n 

d2i = M1 (G).

i=1

On the other hand, M1 (G) = n(n − 1)2 − 4m(n − 1) + M1 (G). Thus,   jT A(G)2 + A(G)2 j = jT A(G)2 j + jT A(G)2 j = M1 (G) + M1 (G) = n(n − 1)2 + 2M1 (G) − 4(n − 1)m. Moreover,   jT A(G) + A(G) j = jT A(Kn )j = n(n − 1).

10

(4)

Therefore, using (1), we have T C(G) + T C(G)   = jT eA(G) + eA(G) j

  1    1 A(G)2 + A(G)2 + A(G)3 + A(G)3 + · · · j = jT 2I + A(G) + A(G) + 2! 3!  1    A(G)2 + A(G)2 j > jT 2I + A(G) + A(G) + 2! = 2n + n(n − 1) + =

1 n(n − 1)2 + M1 (G) − 2(n − 1)m 2

1 n(n2 + 3) + M1 (G) − 2(n − 1)m, 2

and this gives the left inequality of (2). Using Theorem 4 (b) and Lemma 1, we have T C(G) T C(G) ≤ n2 eλ(G)+λ(G) < n2 e1.37n , which gives the right inequality of (3). First we obtain the following lower bound for the total communicability of G. T C(G) = jT eA(G) j 

1 1 = jT I + A(G) + A(G)2 + A(G)3 + · · · j 2! 3!   1 > jT I + A(G) + A(G)2 j 2! = n + 2m +

1 M1 (G). 2

11

By applying this inequality for the graph G and its complement, we get    1 1 T C(G) T C(G) > n + 2m + M1 (G) n + 2m + M1 (G) 2 2   1 1 = n + 2m + M1 (G) n + n(n − 1) − 2m + n(n − 1)2 2 2  1 − 2m(n − 1) + M1 (G) 2 =

 M (G) + n(n2 + 3)  1 1 − m(n − 1) M1 (G) + n2 (n2 + 1) 4 2   + mn (n − 1)2 − 4m .

which gives the required result in the left inequality of (3). By the arithmetic-harmonic-mean inequality, according to the notations in Theorem 6, one can see that M1 (G) ≥

4m2 n .

Hence, using Theorem 6, the following lower bounds

are obtained in terms of n and m. Corollary 7. Let G be a connected graph of order n with m edges and connected complement G. Then T C(G) + T C(G) >

1 4m2 n(n2 + 3) + − 2(n − 1)m 2 n

and T C(G) T C(G) >

  2m 2 n2 (n2 + 1) + m(n − 1) n(n − 1) − 2m + − n + 1 m2 . 2 n

We now give some upper bounds for total communicability of some graph product. Let G and H be two graphs. The Cartesian product, the tensor product and the strong product of these two graphs have the vertex set V (G) × V (H). But they are different in their edge sets. In Cartesian product, G × H, two vertices (u, x) and (v, y) are adjacent iff either [u = v & xy ∈ E(H)] or [x = y & uv ∈ E(G)]. Also, in tensor product G ⊗ H, two vertices (u, x) and (v, y) are adjacent only when [uv ∈ E(G) & xy ∈ E(H)]. And in strong product G  H, two vertices (u, x) and (v, y) are adjacent iff either [u = v & xy ∈ E(H)] or [x = y & uv ∈ E(G)] or [uv ∈ E(G) & xy ∈ E(H)]. In the next result, we obtain an upper bound for each one of these graph products. 12

Theorem 8. Suppose that G and H are connected graphs of orders p and q, respectively. Then (a) T C(G × H) ≤ pq eλ(G)+λ(H) . (b) T C(G ⊗ H) ≤ pq eλ(G)λ(H) . (b) T C(G  H) ≤ pq eλ(G)+λ(H)+λ(G)λ(H) . Furthermore, in each case the equality holds if and only if G and H are regular graphs. Proof. As a consequence of [7, Theorem 2.5.4], we know that if λ1 , . . . , λp and μ1 , . . . , μq are the eigenvalues of the graphs G and H, respectively, then for each 1 ≤ i ≤ p and 1 ≤ j ≤ q, λi + μj are the eigenvalues of G × H and λi μj are the eigenvalues of G ⊗ H and λi + μj + λi μj are the eigenvalues of G  H. Hence, λ(G × H) = λ(G) + λ(H) and λ(G ⊗ H) = λ(G)λ(H) and λ(G  H) = λ(G) + λ(H) + λ(G)λ(H). Thus the first assertion of this theorem is concluded by Theorem 4. Now, to complete the proof, we need to show that G ∗ H is regular if and only if G and H are regular, where ∗ is one of the operations ×, ⊗ or . Here, we denote the degree of the vertex w in a graph K by dK (w). We begin with the Cartesian product G × H. Let (u, v) be a vertex of G × H. We know that dG×H (u, v) = dG (u) + dH (v) (see [22]). Hence, it is obvious that if G and H are regular graphs, then G × H is regular. Conversely, let G × H be a regular graph. If G is not regular (in the case which H is not regular, the proof is similar), then there are two vertices x, y ∈ V (G) such that dG (x) = dH (y). Now, since G × H is regular, for two vertices (x, v), (y, v) ∈ V (G × H) we have dG×H (x, v) = dG×H (y, v) and so dG (x) + dH (v) = dG (y) + dH (v), which is impossible. Therefore the graphs G and H are regular. By a similar method, one can see that the tensor product (resp. strong product) of two graphs G and H is regular if and only if each of G and H is regular. So we omit its proof and the assertion holds. The spectral radius of the graph k K1 ∪

n−k 2

K2 (n − k is even) is obtained in the next

result. Lemma 9. Let G ∼ = k K1 ∪

n−k 2

K2 (n − k is even) be a graph of order n (≥ k). Then √ n − 3 + n2 − 2n + 4k + 1 λ(G) = . 2 13

Proof. One can easily see that the largest eigenvalue of the graph G is given by    λ(G) − k + 1 λ(G) − n + k + 2 = k(n − k), that is, λ(G) =

n−3+

√

n2 − 2n + 4k + 1 . 2

In the next theorem, an upper bound on the spectral radius of a graph G with order n contains k vertices of degree n − 1 is given in terms of n and k. Theorem 10. Let G be a graph of order n with k (1 ≤ k ≤ n) vertices of degree n − 1. Then λ(G) ≤

n−3+

√

n2 − 2n + 4k + 1 2

and equality holds if and only if G ∼ = k K1 ∪

n−k 2

(5)

K2 , where (n − k) is even.

Proof. If G ∼ = Kn , then we have λ(G) = n − 1 and hence the equality holds in (5). So assume that G  Kn . Let y = (y1 , y2 , . . . , yn )T be an eigenvector corresponding to the eigenvalue λ(G) and so A(G)y = λ(G)y. Let S = {v1 , v2 , . . . , vk } (k ≥ 1) be the set of vertices of degree n − 1 in G. Claim 1. yi = yj for each i and j, 1 ≤ i < j ≤ k. Proof of Claim 1. For any vi , vj ∈ S, we have λ(G)yi = 



n 

y

and

λ(G)yj =

=1, =i

n 

y .

=1, =j

Thus λ(G) + 1 (yi − yj ) = 0, that is, yi = yj for 1 ≤ i < j ≤ k as λ(G) > 0 and this 

complete the proof of the Claim 1.

Since G  Kn , so k ≤ n − 2. Let yp = max{y : v ∈ V (G)\S}. For vp ∈ V (G),    y ≤ ky1 +(n−k−2)yp , that is, λ(G)−n+k+2 yp ≤ ky1 . (6) λ(G)yp = v : v vp ∈E(G)

Moreover, λ(G)y1 =

n 

y ≤ (k − 1)y1 + (n − k)yp ,

 that is,

=2

14

 λ(G) − k + 1 y1 ≤ (n − k)yp . (7)

Since y1 , yp > 0, the previous two inequalities yield that    λ(G)−n+k+2 λ(G)−k+1 ≤ k(n−k), that is, λ(G)2 −(n−3)λ(G)−n−k+2 ≤ 0. Since λ(G) > 0, so λ(G) ≤

n−3+

√

n2 − 2n + 4k + 1 2

and the first part of the proof is done. Suppose that equality holds in (5). Then all inequalities in the previous arguments must be equalities. In particular, by the equality in (7), we get yk+1 = yk+2 = · · · = yn , that is, yr = yt for any vr , vt ∈ V (G)\S. Also by the equality in (6), we get dp = n − 2. For any vr ∈ V (G)\S, 

y = λ(G)yr = λ(G)yp =

v : v vr ∈E(G)





y , that is,

v : v vp ∈E(G)

v : v vr ∈E(G) v ∈V (G)\S

y =

 v : v vp ∈E(G) v ∈V (G)\S

Since yk+1 = yk+2 = · · · = yn , so dk+1 = dk+2 = · · · = dn = n − 2. Also since 2m = k(n − 1) + (n − k)(n − 2) = n(n − 1) − (n − k), it concludes that n − k is even. Hence G ∼ = k K1 ∪ n−k K2 (n − k is even). 2

Conversely, the equality holds in (5) for the graph k K1 ∪

n−k 2

K2 (n − k is even), by

Lemma 9. Corollary 11. Let G ( Kn ) be a graph of order n. Then √ n − 3 + n2 + 2n − 7 λ(G) ≤ 2

(8)

and equality holds if and only if G ∼ = Kn − e (e is any edge in Kn ). Proof. Let k be the number of vertices of degree n − 1 in G. Then 0 ≤ k ≤ n − 2 as G  Kn . If k = 0, then the maximum degree Δ ≤ n − 2 and hence √ n − 3 + n2 + 2n − 7 λ(G) ≤ Δ ≤ n − 2 < . 2 Otherwise, 1 ≤ k ≤ n − 2. Then by Theorem 10, we have √ n − 3 + n2 − 2n + 4k + 1 λ(G) ≤ . 2 15

y .

Since

√

n2 − 2n + 4x + 1 (1 ≤ x ≤ n − 2) 2 is an increasing function on [1, n − 2], one can easily see that √ √ n − 3 + n2 + 2n − 7 n − 3 + n2 − 2n + 4k + 1 = f (k) ≤ f (n − 2) = . λ(G) ≤ 2 2 f (x) =

n−3+

Also, the equality holds in (8) if and only if G ∼ = k K1 ∪ k = n − 2, that is, G ∼ = Kn − e (e is any edge in Kn ).

n−k 2

K2 (n − k is even) and

Recall the definition of the total communicability T C of a graph. It counts the sum of the total number of walks between each two vertices, weighted in decreasing order of their length by a factor

1 k! .

Thus it is considering shorter walks more influential

than longer one and it seems that the graphs with the maximum number of short walks have the maximum T C. The next results, also, confirm this fact for the total communicability of a graph. As instance, it says among the graphs with n vertices, the complete graph Kn has the maximum T C. Theorem 12. Let G ( Kn ) be a graph of order n with k (≥ 1) vertices of degree n − 1. Then T C(G) ≤ n ep < n en−1 = T C(Kn ) where

√

n2 − 2n + 4k + 1 . 2 Moreover, T C(G) = n ep if and only if G ∼ = n2 K2 (n is even). p=

n−3+

Proof. Since G  Kn , by Theorems 4 (b) and 10, we get the required result. Moreover, T C(G) = n ep if and only if G is regular and G ∼ = k K1 ∪ n−k K2 (n − k is even) with k ≤ n − 2, that is, G ∼ =

n 2

2

K2 (n is even).

Corollary 13. Let G be a graph of order n. (a) T C(G) ≤ nen−1 and equality holds if and only if G is the complete graph Kn . √

(b) If G is a tree, then T C(G) ≤ ne

n−1

and the upper bound occurs only when G is

the path P2 .

16

3

(c) If G is a unicyclic graph, then T C(G) ≤ neλ(Sn ) , and the upper bound occurs only when G is the cycle C3 . Proof. In [6] it is proved that λ(G) ≤ λ(Kn ) and the equality holds if and only if ∼ Kn . Now, since the function ez is strictly increasing, by Theorem 4, T C(G) ≤ G= neλ(G) ≤ neλ(Kn ) and the equality holds if and only if λ(G) = λ(Kn ) = n − 1 and G is (n − 1)-regular, which completes the proof of (a). In [6], it is shown that among the trees with n vertices, the star K1,n−1 has the √ maximum spectral radius. So, if G is a tree, then λ(G) ≤ λ(K1,n−1 ) = n − 1 and the equality holds only when G is the star K1,n−1 . Now, since ez is a strictly increasing √

function, by Theorem 4 we have T C(G) ≤ neλ(G) ≤ ne n−1 and the equality holds √ √ only when λ(G) = n − 1 and the tree G is ( n − 1)-regular. Since each tree has at least two vertices of degree 1, so G is the path P2 and the proof of (b) is completed. Now, suppose that G is a unicyclic graph. Then λ(G) ≤ λ(Sn3 ) with equality if and 3

only if G = Sn3 (see [17]). So, using Theorem 4, T C(G) ≤ neλ(Sn ) . Also, the equality holds only when G = Sn3 and Sn3 is regular. Thus Sn3 ∼ = C3 , as desired.

In the next result, we obtain some upper bounds for total communicability of a graph in terms of different graph parameters. Corollary 14. Let G be a graph with n vertices and m edges. (1) T C(G) ≤ ne Kn . (2) T C(G) ≤ ne

√

2m−n+1

√

δ(G)−1+

and the equality holds if and only if G is the complete graph

(δ(G)+1)2 +4(2m−nδ(G)) 2

.

  dG (vi )dG (vj ) | 1 ≤ i, j ≤ n and vi vj ∈ E(G) . (3) T C(G) ≤ nep , where p = max

(4) T C(G) ≤ neq , where q = max



√

mi mj | 1 ≤ i, j ≤ n and vi vj ∈ E(G)



is the average degree of the vertices adjacent to vi ∈ V (G). √ (5) T C(G) ≤ ne 2m−(n−1)δ(G)+(δ(G)−1)Δ(G) . Moreover, from (2) − (5) the equality holds if and only if G is a regular graph.

17

and mi

Proof. Since the function ez is strictly increasing, by Theorem 4 and the following results we get the assertions. (1) (Hong [18]) λ(G) ≤

√

2m − n + 1, where the equality holds iff G is one of the

graphs Kn or K1,n−1 . (2) (Hong et al. [19]) λ(G) ≤

δ(G)−1+

√

(δ(G)+1)2 +4(2m−nδ(G)) , 2

where the equality

holds iff G is either a regular graph or a bidegreed graph in which each vertex is of degree either δ(G) or n − 1. (3) (Berman and Zhang [3]) λ(G) ≤ max

  di dj | 1 ≤ i, j ≤ n and vi vj ∈ E(G) ,

where the equality holds iff G is a regular or bipartite semiregular graph.

 √ (4) (Das and Kumar [9]) λ(G) ≤ max mi mj | 1 ≤ i, j ≤ n and vi vj ∈ E(G) and the equality holds only when G is either a graph with all the vertices of equal average degree or a bipartite graph with vertices of same set having equal average degree. (5) (Das and Kumar [9]) λ(G) ≤



2m − (n − 1)δ(G) + (δ(G) − 1)Δ(G) and the

equality holds if and only if G is a regular graph or a star graph.

4

Computational Results

In this section, we carry out some computations for variety of connected graphs, with the aim of comparing the bounds for total communicability. In particular, we are interested in comparing the lower bounds for total communicability of a connected  ω(G) graph G, such as LT C(G) = ω(G)−1 λ(G)eλ(G) and the Estrada index EE(G), which are obtained in Theorem 4 and [2, Proposition 1], respectively. Also, we would like to compare the upper bounds in Corollary 14 to show that for some graphs, an upper bound can be better than the other. Here, the i-th upper bound in Corollary 14 for a graph G is denoted by Ui (G), i = 1, . . . , 5. The values of T C(G), the various mentioned upper bounds in Corollary 14 and the lower bounds LT C(G), EE(G) for the graphs shown in Figure 1, are given in the Table 1. 18

Figure 1: The graphs Gi , (1 ≤ i ≤ 5). Table 1: Total communicability of graphs in Figure 1 and their upper and lower bounds. G T C(G) U1 (G) U2 (G) U3 (G) U4 (G) U5 (G) EE(G) LT C(G) G1

47.3020

57.9121

57.9122

84.5941

100.4277

57.9122

13.5437

36.5905

G2

95.3364

165.4029

120.5132

191.6865

153.2591

141.7461

21.1218

68.7343

G3

64.5035

120.5132

77.7355

69.4946

73.0950

84.5642

16.2630

52.2689

G4

186.3293

288.5138

243.0632

327.5889

234.7277

253.0069

36.3523

166.8268

As we can see in the Table 1, each upper bound in Corollary 14 for some graphs is better than the other ones. For instance, the upper bounds U2 , U3 and U4 are the best upper bounds for the graphs G2 , G3 and G4 , respectively. Also, the upper bounds U1 and U5 for the graph G1 are better than the upper bounds U3 and U4 . Also, in the two last columns, one can compare the lower bounds LT C and EE, for the graphs Gi ’s, i = 1, . . . , 4, which shows that LT C is a better lower bound for total communicability of these graphs compared to the Estrada index. In the sequel, we have calculated the parameters T C(G) − LT C(G) and T C(G) − EE(G) for all 273193 connected graphs with at most 9 vertices and depicted their frequency plot in Figure 2. Note that in this figure, horizontal axis shows the values of T C(G) − LT C(G) and T C(G) − EE(G), and vertical axis shows the frequency of these values. One can see that the frequency of lower values of errors for LT C is higher than EE, so LT C is a better lower bound for T C compared to EE for the connected graphs up to 9 vertices. C For further investigating of the upper bounds in Corollary 14, the mean of ( UiT−T C ) for all connected graphs of order n, n = 3, . . . , 9, has been computed in the Table 2

19

Figure 2: The frequency of T C − LT C (red) and T C − EE (blue) for the connected graphs with at most 9 vertices. and shown in Figure 3, where i = 1, . . . , 5. According to these data, it seems that U4 is a better upper bound for T C(G) when the order of G grows. Table 2: The relative errors of the i-th upper bound (Ui ) in Corollary 14 for all connected graphs of order n. n

No. Connected Graphs

C mean( U1T−T C )

C mean( U2T−T C )

C mean( U3T−T C )

C mean( U4T−T C )

C mean( U5T−T C )

3 4 5 6 7 8 9

2 6 21 112 853 11117 261080

0.0138 0.1244 0.2935 0.5317 0.8317 1.2164 1.6945

0.0138 0.0652 0.1688 0.3230 0.5238 0.7488 0.9927

0.0138 0.2633 0.5915 1.0394 1.5709 2.1811 2.8785

0.0138 0.1369 0.2884 0.4321 0.5765 0.6855 0.7699

0.0138 0.0800 0.2333 0.4103 0.6500 0.9167 1.2199

Acknowledgement The authors are much grateful to the anonymous referee for his/her valuable comments on our paper, which have considerably improved the presentation of this paper. The 20

Figure 3: The relative errors of the i-th upper bound (Ui ) in Corollary 14 for all connected graphs of order n. first author was supported by the Sungkyun research fund, Sungkyunkwan University, 2017, and National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642. The second and the third author were supported by National Elites Foundation under Grant Numbers 15/89682 and 15/93173, respectively.

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