On the distance α-spectral radius of a connected graph

For a connected graph G and α∈[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \in [0,1)$\end{document}, the distance α-spectral radius of G is the spectral radius of the matrix Dα(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D_{\alpha }(G)$\end{document} defined as Dα(G)=αT(G)+(1−α)D(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D_{\alpha }(G)=\alpha T(G)+(1-\alpha )D(G)$\end{document}, where T(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T(G)$\end{document} is a diagonal matrix of vertex transmissions of G and D(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D(G)$\end{document} is the distance matrix of G. We give bounds for the distance α-spectral radius, especially for graphs that are not transmission regular, propose local graft transformations that decrease or increase the distance α-spectral radius, and determine the graphs that minimize and maximize the distance α-spectral radius among several families of graphs.


Introduction
We consider simple and undirected graphs. Let G be a connected graph of order n with vertex set V (G) and edge set E(G). For u, v ∈ V (G), the distance between u and v in G, denoted by d G (u, v) or simply d uv if the graph G is clear from the context, is the length of a shortest path from u to v in G. The distance matrix of G is the n × n matrix D(G) = (d G (u, v)) u,v∈V (G) . For u ∈ V (G), the transmission of u in G, denoted by T G (u), is defined as the sum of distances from u to all other vertices of G, i.e., T G (u) = v∈V (G) d G (u, v). The transmission matrix T(G) of G is the diagonal matrix of transmissions of G. Then Q(G) = T(G)+D(G) is the distance signless Laplacian matrix of G, proposed recently in [1]. Arisen from a data communication problem, the spectrum of the distance matrix was studied by Graham and Pollack [12] in 1971, early related work may be found also in [10,11], and now it has been studied extensively, see the recent survey [2] and the very recent papers [4,5,17,18,26]. The distance signless Laplacian spectrum has also received much attention, see, e.g., [1,3,4,7,15,16,29].
In this paper, we give sharp bounds for the distance α-spectral radius, and particularly an upper bound for the distance α-spectral radius of connected graphs that are not transmission regular, and propose some types of graft transformations that decrease or increase the distance α-spectral radius. We also determine the unique graphs with minimum distance α-spectral radius among trees and unicyclic graphs, respectively, as well as the unique graphs (trees) with maximum and second maximum distance α-spectral radii, and the unique graph with maximum distance α-spectral radius among connected graphs with given clique number, and among odd-cycle unicyclic graphs, respectively.

Preliminaries
Let G be a connected graph with V (G) = {v 1 , . . . , v n }. A column vector x = (x v 1 , . . . , x v n ) ∈ R n can be considered as a function defined on V (G) which maps vertex v i to x v i , i.e., x(v i ) = x v i for i = 1, . . . , n. Then or equivalently, Since D α (G) is a nonnegative irreducible matrix, by the Perron-Frobenius theorem, μ α (G) is simple and there is a unique positive unit eigenvector corresponding to μ α (G), which is called the distance α-Perron vector of G. If x is the distance α-Perron vector of G, then for each u ∈ V (G), which is called the α-equation of G at u. For a unit column vector x ∈ R n with at least one nonnegative entry, by Rayleigh's principle, we have μ α (G) ≥ x D α (G)x with equality if and only if x is the distance α-Perron vector of G.
As in [27], we have the following result.

Lemma 2.1
Suppose that G is a connected graph, η is an automorphism of G, and x is the distance α-Perron vector of G. Then for u, v ∈ V (G), η(u) = v implies that x u = x v .
Proof Let P = (p uv ) u,v∈V (G) be the permutation matrix such that p vu = 1 if and only if η(u) = v for u, v ∈ V (G). We have D α (G) = P D α (G)P and Px is a positive unit vector. Thus implying Px is also the distance α-Perron vector of G. Thus Px = x, and the result follows.
Let G be a graph. For v ∈ V (G), let N G (v) be the set of neighbors of v in G, and deg G (v) be the degree of v in G. Let Gv be the subgraph of G obtained by deleting v and all edges containing v. For S ⊆ V (G), let G[S] be the subgraph of G induced by S. For a subset E of E(G), G -E denotes the graph obtained from G by deleting all the edges in E , and in particular, For a subset E of E(G), denote G + E the graph obtained from G by adding all edges in E , and in particular, For a nonnegative square matrix A, the Perron-Frobenius theorem implies that A has an eigenvalue that is equal the maximum modulus of all its eigenvalues; this eigenvalue is called the spectral radius of A, denoted by ρ(A). Note that μ α (G) = ρ(D α (G)) for a connected graph G. Restating The transmission of a connected graph G, denoted by σ (G), is the sum of distances between all unordered pairs of vertices in G. Clearly, σ (G) = 1 2 v∈V (G) T G (v). A graph is said to be transmission regular if T G (v) is a constant for each v ∈ V (G). By Rayleigh's principle, we have Lemma 2.4 Suppose that G is a connected graph of order n. Then μ α (G) ≥ 2σ (G) n with equality if and only if G is transmission regular.
For an n × n nonnegative matrix A = (a ij ), let r i be the ith row sum of A, i.e., r i = n j=1 a ij for i = 1, . . . , n, and let r min and r max be the minimum and maximum row sums of A, respectively.
Moreover, the first equality holds if r ia iq = m and a iq = t for all i ∈ S \ {q}, and the second equality holds if r ia ip = and a ip = s for all i ∈ S \ {p}.
Let J s×t be the s × t matrix of all 1's, 0 s×t the s × t matrix of all 0's, and I s the identity matrix of order s.
Let K n , P n , and S n be the complete graph, the path, and the star of order n, respectively. Let C n denote the cycle of order n ≥ 3.
For a connected graph G, let T min (G) and T max (G) be the minimum and maximum transmissions of G, respectively.

Bounds for the distance α-spectral radius
Let G be a connected graph of order n. Note that D α (K n ) = α(n -1)I n + (1α)(J n×n -I n ), and thus μ α (K n ) = n -1. By Lemma 2.3, we have μ α (G) ≥ n -1 with equality if and only if G ∼ = K n .
If (d 1 , . . . , d n ) is the nonincreasing degree sequence of a graph G of order at least 2, then d 1 (resp. d 2 ) is the maximum (resp. second maximum) degree, d n (resp. d n-1 ) is the minimum (resp. second minimum) degree of G. The diameter of G is the maximum distance between all vertex pairs of G. Using techniques from [33] by considering the first two minima or maxima of the entries of the distance α-Perron vector, we may prove the following lower and upper bounds: If G is a connected graph of order n ≥ 2 with maximum degree and second maximum degree , then with equality if and only if G is regular with diameter at most 2. If G is a connected graph of order n ≥ 2 with minimum degree δ and second minimum degree δ , then . The proof of the above bounds may be found in the early version of this paper at arXiv:1901.10180.
Similarly, bounds for the distance α-spectral radius for connected bipartite graphs may be obtained as in [33].
A connected graph G of order n is distinguished vertex deleted regular (DVDR) if there is a vertex v of degree n -1 such that Gv is regular. By the techniques in [3], we have the following bounds. For completeness, we include a proof here.

The first equality holds if and only if G is a complete graph and the second equality holds if and only if G is a DVDR graph.
Proof Let M be the submatrix of D α (G) obtained by deleting the row and column corresponding to vertex v. Let M be the matrix obtained from M by reducing some nondiagonal entries of each row with row sum greater than m 2 in M such that M is nonnegative and each row sum in M is m 2 .
Conversely, if G is a complete graph, then it is obvious that the lower bound for μ α (G) is attained.
Let C be the submatrix of D α (G) obtained by deleting the row and column corresponding to vertex u. Let C be the matrix obtained from C by adding positive numbers to nondiagonal entries of each row with row sum less than m 1 in C such that each row sum in C is m 1 . Let D (2) be the matrix obtained from D α (G) by replacing all (w, u)-entries by (1α)e(u) for w ∈ V (G) \ {u}, and replacing the submatrix C by C . Note that D α (G) and D (2) are nonnegative and irreducible, and D (2) ≥ D α (G). By Lemma 2.2, μ α (G) ≤ ρ(D (2) ) with equality if and only if D α (G) = D (2) . By applying Lemma 2.5 to D (2) , we obtain the upper bound for μ α (G).
Suppose that this upper bound is attained. By Lemma 2.2, D α (G) = D (2) . As all (w, u)- Recall from [3] that an incomplete connected graph of order n is a DVDR graph if and only if except one vertex of degree n -1 each other vertex has the same transmission. Thus, the upper bound for μ α (G) is attained if and only if G is a DVDR graph.
We mention that more bounds for μ α (G) may be derived even from some known bounds for nonnegative matrices, see, e.g., [9].
Let G be a connected graph of order n. Let Λ = T max (G). As μ α (G) ≤ Λ with equality if and only if G is transmission regular. For a connected non-transmission-regular graph G of order n, Liu et al. [19] showed that Note that 4σ (G) < n 2 Λ. We show new bounds as follows: Instead of proving the two inequalities, we prove the following somewhat general result.

Theorem 3.2 Let G be a connected non-transmission-regular graph of order n. Then
We need to estimate {w,z}⊆V (G) d wz (x wx z ) 2 . Let P = w 0 w 1 . . . w be a shortest path connecting u and v, where w 0 = u, w = v, and ≥ 1. Obviously, and thus For 1 ≤ i ≤ -1 and ≥ 2, by the Cauchy-Schwarz inequality, we have and thus Case 1. u and v are adjacent, i.e., = 1.
In this case, we have Viewed as a function of Case 2. u and v are not adjacent, i.e., ≥ 2. Suppose first that is even. Then Viewed as a function of i.e., As a function of , the expression on the right-hand side in the above inequality is strictly increasing for ≥ 2. Thus we have Now suppose that is odd. Then Thus, as early, we have (1α)( 2 + 4 + 4n -5)(nΛ -2σ (G)) 8(nΛ -2σ (G)) + (1α)( 2 + 4 + 4n -5) As a function of , the expression on the right-hand side in the above inequality is strictly increasing for ≥ 3. Thus we have The result follows by combining Cases 1 and 2.

Effect of graft transformations on distance α-spectral radius
In this section, we study the effect of some local graft transformations on distance αspectral radius.
A path u 0 · · · u r (with r ≥ 1) in a graph G is called a pendant path (of length r) at u 0 if deg G (u 0 ) ≥ 3, the degrees of u 1 , . . . , u r-1 (if any exists) are all equal to 2 in G, and deg G (u r ) = 1. A pendant path of length 1 at u 0 is called a pendant edge at u 0 .
A vertex of a graph is a pendant vertex if its degree is 1. A cut edge of a connected graph is an edge whose removal yields a disconnected graph.
If P is a pendant path of G at u with length r ≥ 1, then we say G is obtained from H by attaching a pendant path P of length r at u with H = G[V (G) \ (V (P) \ {u})]. If the pendant path of length 1 is attached to a vertex u of H, then we also say that a pendant vertex is attached to u.
Proof Let G 1 and G 2 be the components of Guv containing u and v, respectively. Let x be the distance α-Perron vector of G uv . By Lemma 2.1, x u = x v . As we pass from G to G uv , the distance between a vertex in V (G 1 ) \ {u} and a vertex in V (G 2 ) is decreased by 1, the distance between a vertex V (G 1 ) \ {u} and u is increased by 1, and the distances between all other vertex pairs remain unchanged. Thus implying μ α (G)μ α (G uv ) > 0, i.e., μ α (G) > μ α (G uv ).
Proof Let x be the distance α-Perron vector of G.
As we pass from G to G , the distance between a vertex in V (G 2 ) and a vertex in V K is increased by d G (u, v ), the distance between a vertex w in V (G 1 ) \ {u} and a vertex in V K is decreased by d G (w, u)d G (w, v ), which is at most d G (u, v ), and the distances between all other vertex pairs are increased or remain unchanged. Thus . Suppose that Γ < 0. As we pass from G to G , the distance between a vertex in V (G 1 ) and a vertex in V K is increased by d G (u, v ), the distance between a vertex w in V (G 2 ) \ {u} and a vertex in V K is decreased by d G (w, u)d G (w, v ), which is at most d G (u, v ), and the distances between all other vertex pairs are increased or remain unchanged. Thus Weak versions of previous theorem for α = 0 have been given in [28,30] and a weak version for α = 1 2 may be found in [16]. For positive integer p and a graph G with u ∈ V (G), let G(u; p) be the graph obtained from G by attaching a pendant path of length p at u. Let G(u; 0) = G, and in this case a pendant path of length 0 is understood the trivial path consisting of a single vertex u.
For nonnegative integers p, q and a graph G, let G u (p, q) be the graph H(u; q) with H = G(u; p). The following corollary has been known for α = 0 in [24,28] and α = 1 2 in [15,16].

Theorem 4.3 Suppose that G is a connected graph with three edge-disjoint induced sub-
Proof Let x be the distance α-Perron vector of G. Let As we pass from G to G , the distance between a vertex in V (G 2 ) and a vertex in V (G 3 ) \ {u, v} is increased by 1, the distance between a vertex in V (G 1 ) and a vertex in V (G 3 ) \{u, v} may be increased, unchanged, or decreased by 1, and the distances between any other vertex pairs remain unchanged. Thus a contradiction. Thus, if Γ ≥ 0, then μ α (G) < μ α (G ). Suppose that Γ < 0. As earlier, we have and thus μ α (G) < μ α (G ).
A weak version of previous theorem for α = 1 2 has been established in [16]. For nonnegative integers p, q and a graph G with u, v ∈ V (G), let G u,v (p, q) be the graph H(v; q) with H = G(u; p). The following corollary has been known for α = 0, 1 2 in [15,32].

Graphs with small or large distance α-spectral radius
First we determine the graphs with minimum distance α-spectral radius among trees and unicyclic graphs. Proof The result is trivial if n = 1, 2, 3. Suppose that n ≥ 4. Let G be a tree of order n such that μ α (G) is as small as possible. Let d be the diameter of G.
For n -1 ≥ 3 and 1 ≤ a ≤ n-2 2 , let D n,a be the tree obtained from vertex-disjoint S a+1 with center u and S n-a-1 with center v by adding an edge uv. Let T be a tree of order n with minimum distance α-spectral radius, where T S n . Let d be the diameter of T. Then d ≥ 3. Suppose that d ≥ 4. Let v 0 v 1 · · · v d be a diametral path of T. Note that T v 1 v 2 S n . By Theorem 4.1, μ α (T v 1 v 2 ) < μ α (T), a contradiction. Thus d = 3, implying T ∼ = D n,a for some a with 1 ≤ a ≤ n-2 2 .
Let S + n is the graph obtained from S n by adding an edge between two vertices of degree one.

Lemma 5.1 ([29])
Let G be a unicyclic graph of order n ≥ 6. If G S + n , then Note that for n = 5, we have σ (C n ) = σ (S + n ). So, in the above lemma, the condition n ≥ 6 is necessary. Proof Suppose that G S + n . We only need to show that μ α (G) > μ α (S + n ). By Lemmas 2.4 and 5.1, we have By [20, p. 24, Theorem 1.1] or by Theorem 3.2, we have Since n ≥ 8, we have as desired.
The result in Theorem 5.2 for α = 0, 1 2 has been known in [29,31]. In the following, we determine the graphs with maximum distance α-spectral radius among some classes of graphs.
For 2 ≤ ≤ n -1, let B n, be a tree obtained by attaching -1 pendant vertices to a terminal vertex of the path P n-+1 . In particular, B n,2 = P n and B n,n-1 = S n . The following theorem for α = 0, 1 2 was given in [16,24] for trees. Proof Let G be a graph among connected graphs of order n with maximum degree such that μ α (G) is as large as possible. Then G has a spanning tree T with maximum degree . By Lemma 2.3, μ α (G) ≤ μ α (T) with equality if and only if G ∼ = T. Thus G is a tree. The result is trivial if n = 3, 4 and if = 2, n -1. Suppose that 3 ≤ ≤ n -2. We only need to show that G ∼ = B n, .
Let u ∈ V (G) with deg G (u) = . Suppose that there exists a vertex different from u with degree at least 3. Then we may choose such a vertex w of degree at least 3 such that d G (u, w) is as large as possible. Obviously, there are two pendant paths, say P and Q, at w of lengths at least 1. Let p and q be the lengths of P and Q, respectively. Assume that p ≥ q. Let H w (p, q). Note that G = H w (p + 1, q -1) is a tree of order n with maximum degree . By Corollary 4.1, μ α (G) < μ α (G ), a contradiction. Then u is the unique vertex of G with degree at least 3, and thus G consists of pendant paths, say Q 1 , . . . , Q at u. If two of them, say Q i and Q j with i = j are of lengths at least 2, then G ∼ = H u (r, s) , and r and s are the lengths of Q i and Q j , respectively. Assume that r ≥ s. Obviously, G = H u (r + 1, s -1) is a tree of order n with maximum degree . By Corollary 4.1, μ α (G) < μ α (G ), also a contradiction. Thus there is exactly one pendant path at u of length at least 2, implying G ∼ = B n, .
If G is a connected graph of order 1 or 2, then G ∼ = P n . If G is a connected graph of order 3, then G ∼ = P 3 , K 3 , and by Lemma 2.3, μ α (K 3 ) < μ α (P 3 ).
Ruzieh and Powers [23] showed that P n is the unique connected graph of order n with maximum distance 0-spectral radius, and it was proved in [25] that B n,3 is the unique tree of order n different from P n with maximum distance 0-spectral radius. For α = 1 2 , the following theorem was given in [16]. Proof First suppose that G is a tree. If n = 4, then the result follows from Theorem 4.1. Suppose that n ≥ 5. Let be the maximum degree of G. Since G P n , we have ≥ 3. By Theorem 5.3, μ α (G) ≤ μ α (B n, ) with equality if and only if G ∼ = B n, . By Corollary 4.1, μ α (G) ≤ μ α (B n, ) ≤ μ α (B n,3 ) < μ α (P n ) with equalities if and only if = 3 and G ∼ = B n, , i.e., G ∼ = B n, 3 . Now suppose that G is not a tree. Then G contains at least one cycle. If there is a spanning tree T with T P n , then by Lemma 2.3 and the above argument, we have μ α (G) < μ α (T) ≤ μ α (B n,3 ). If any spanning tree of G is a path, then G is a cycle C n . Now we only need to show that μ α (C n ) < μ α (B n, 3 ).
Let C n = u 1 u 2 · · · u n u 1 and T = C n -{u 1 u 2 , u 2 u 3 } + u 2 u n . Then T ∼ = B n, 3 . Let x be the distance α-Perron vector of C n . By Lemma 2.3, we have x u 1 = · · · = x u n . As we pass from C n to T , the distance between u 2 and u 1 is increased by 1, the distance between u 2 and u i with 3 ≤ i ≤ n+1 2 is increased by n-2i+3, the distance between u 2 and u i with n+1 2 +2 ≤ i ≤ n is decreased by 1, and the distances between all other vertex pairs are increased or remain unchanged. Thus (n -2i + 3) = 2x 2 u 1 1 + n -1 - and therefore μ α (C n ) < μ α (B n,3 ), as desired.
A clique of G is a subset of vertices whose induced subgraph is a complete graph, and the clique number of G is the maximum number of vertices in a clique of G. For 2 ≤ ω ≤ n. Let Ki n,ω be the graph obtained from a complete graph K ω and a path P n-ω by adding an edge between a vertex of K ω and a terminal vertex of P n-ω if ω < n and let Ki n,ω = K n if ω = n. In particular, Ki n,2 ∼ = P n for n ≥ 2. The following result for α = 0, 1 2 was given in [15,21]. Proof It is trivial if ω = n and it follows from Theorem 5.4 if ω = 2. Suppose that 3 ≤ ω ≤ n -1. Let G be a graph among connected graphs of order n with clique number ω such that μ α (G) is as large as possible. We only need to show that G ∼ = Ki n,ω .
Let Recall that Ki n,3 is the unique unicyclic graph of order n ≥ 3 with maximum distance 0-spectral radius [31], and the unique odd-cycle unicyclic graph of order n ≥ 3 with maximum distance 1 2 -spectral radius [15]. Proof If n = 3, 4, the result is trivial. Suppose that n ≥ 5. Let G be a graph with maximum distance α-spectral radius among unicyclic odd-cycle graphs of order n. We only need to show that G ∼ = Ki n, 3 .
Note that the length of C is odd. As we pass from G to G , the distance between a vertex in S 1 and a vertex in S 3 is increased by at least 1, the distance between S 2 and V (T 2k+1 ) is decreased by 1, and the distance between all other vertex pairs are increased or remain unchanged. Thus If Γ ≥ 0, then μ α (G ) > μ α (G), a contradiction. Thus Γ < 0. Let G = G-v 2k v 2k-1 +v 2k v 1 . As we pass from G to G , the distance between a vertex in S 1 and a vertex in U 2 is increased by at least 1, the distance between U 3 and V (T 2k ) is decreased by 1, and the distance between all other vertex pairs are increased or remain unchanged. As above, we have Thus μ α (G ) > μ α (G), also a contradiction. It follows that k = 1, i.e., the unique cycle of G is of length 3. Obviously, T i is a tree for 1 ≤ i ≤ 3. For 1 ≤ i ≤ 3, by Corollary 4.1, if T i is nontrivial, then it is a path with a terminal vertex v i . Then by Corollary 4.2, only one T i is nontrivial. Thus G ∼ = Ki n,3 .
Let G be a unicyclic graph of order n ≥ 4 with maximum distance α-spectral radius. By Corollary 4.1, the maximum degree of G is 3 and all vertices of degree 3 lie on the unique cycle. Let u be a vertex of degree 3 and P be the pendant path at u. Let v and w be the two neighbors of u on the cycle, and z the neighbor of u on P. Let G 1 = Guw + vw and G 2 = Guw + wz. Then μ α (G) < max{μ α (G 1 ), μ α (G 2 )} if the length of the cycle of G is odd, see [4,Lemma 6.11]. Note that the argument does not work when the length of the cycle of G is even. So we need other ways to determine the unicyclic graph(s) with maximum distance α-spectral radius even for α = 1 2 .

Remarks
In this paper, we study the distance α-spectral radius of a connected graph. We consider bounds for the distance α-spectral radius, local transformations to change the distance α-spectral radius, and the characterizations for graphs with minimum and/or maximum distance α-spectral radius in some classes of connected graphs. Besides the distance α-spectral radius, we may concern other eigenvalues of D α (G) for a connected graph G. We give examples.
Using these inequalities, and as in the recent work of Atik and Panigrahi [3], we have Theorem 6.1 Let G be a connected graph and λ be any eigenvalue of D α (G) other than the distance α-spectral radius. Then For matrix A, we have λ 1 (A) = αT min (G) + (1α)(n -1) and λ j (A) = αT min (G) -1 + α for j = 2, . . . , n. Thus, for j = 2, . . . , n, we have by the above Weyl's inequalities that This completes the proof.
Let G be a connected graph and λ be any eigenvalue of D α (G) other than the distance α-spectral radius. By previous theorem, we have |λ| ≤ T max (G) -(1α)(n -2).
The distance α-energy of a connected graph G of order n is defined as Then E 0 (G) is the distance energy of G [14,33], while is half of the distance signless Laplacian energy of G [8]. Thus, it is possible to study the distance energy and the distance signless Laplacian energy in a unified way.