ON GRAFT TRANSFORMATIONS DECREASING DISTANCE SPECTRAL RADIUS OF GRAPHS

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. In this paper, we give several less restricted graft transformations that decrease the distance spectral radius, and determine the unique graph with minimum distance spectral radius among homeomorphically irreducible unicylic graphs on n ≥ 6 vertices, and the unique tree with minimum distance spectral radius among trees on n vertices with given number of vertices of degree two, respectively. Mathematics Subject Classification. 05C50, 15A48. Received November 21, 2020. Accepted May 27, 2021.

In this paper, we propose some graft transformations in a more elaborate way. We propose some graft transformations with less restricted conditions that decrease the distance spectral radius, and as applications, we identify the unique graphs that minimize the distance spectral radius among homeomorphically irreducible unicylic graphs on n ≥ 6 vertices, and among trees on n vertices with given number of vertices of degree two, respectively.

Preliminaries
Let G be a connected graph with V (G) = {v 1 , . . . , v n }. Since D(G) is irreducible, by Perron-Frobenius theorem, ρ(G) is simple and there is a unique unit positive eigenvector corresponding to ρ(G), which is called the distance Perron vector of G, denoted by x(G). If y = (y v1 , . . . , y vn ) ∈ R n is unit and has at least one nonnegative entry, then by Rayleigh's principle, we have ρ(G) ≥ y T D(G)y with equality if and only if y = x(G). If x = x(G), then for each u ∈ V (G), we have ρ(G)x u = v∈V (G) d G (u, v)x v , which is called the distance eigenequation of G at u.
Let N ⊆ (N G (u) \ N G (v)) \ {v}. Let G = G − uw + vw for w ∈ N . We say that G is obtained from G by moving edge uw at u from u to v.
For a connected graph G with V 1 ⊆ V (G), let σ G (V 1 ) be the sum of the entries of the distance Perron vector of G corresponding to the vertices in V 1 . Furthermore, if all the vertices of V 1 induce a connected subgraph H of G, then we write σ G (H) instead of σ G (V 1 ).
A component of a graph is a maximal connected subgraph, and a cut edge is an edge of a graph whose removal increases the number of components of the graph.
For a connected graph G, let s(G) be the minimum row sum of D(G). In [17], s(G) n is called the mean vertex deviation of G, where n = |V (G)|. It is known that ρ(G) ≥ s(G), see Theorem 1.1 in page 24 of [10].

Graft transformations that decrease the distance spectral radius
Firstly, we give a result related to (the entries of) the distance Perron vector, which will be frequently used in the subsequent proofs.
Lemma 3.1. Suppose that v, w be two non-adjacent neighbors of vertex u in a connected graph G. Let x = x(G).
Now we turn our attention to some graft transformations that decrease the distance spectral radius.
Theorem 3.2. Let G be a graph consisting of nontrivial connected graphs G 1 and G 2 sharing a unique vertex u such that E(G) = E(G 1 ) ∪ E(G 2 ). Suppose that u has neighbor v of degree at least two in G 2 satisfying that, , z). Let G be the graph obtained from G by moving all the edges at v except uv from v to u. Then ρ(G) > ρ(G ).
. Note first that, as we pass from G to G , the distance between a vertex of S and a vertex of V (G 1 ) is decreased by 1, and the distance between a vertex of S and v is increased by 1. So, to prove the claim, we need only to show that the distance between any other vertex pairs is decreased or remains unchanged.
It is evident that the distance between any two vertices in V (G 1 ) ∪ {v} remains unchanged as we pass from G to G .
Suppose that z 1 , z 2 ∈ V (G 2 ) \ {u, v}. Let P be a path from z 1 to z 2 with length d G (z 1 , z 2 ) in G. If v lies outside P , then P is also a path connecting z 1 and z 2 in G . Suppose that v lies on P . If u lies outside P , then the path obtained from P by replacing v with u is a path connecting z 1 and z 2 in G . Otherwise, u lies on P . In this case, uv appears to be an edge on P . So the path obtained from P by deleting v is a path connecting z 1 and z 2 in G . So the distance between any two vertices in V (G 2 ) \ {u, v} is decreased or remains unchanged as we pass from G to G .
Suppose that z ∈ S : Then v lies outside P , so P is also a path from u to z in G . Therefore, the distance between a vertex in S and a vertex in V (G 1 ) ∪ {v} is decreased or remains unchanged as we pass from G to G . Now we complete the proof of the claim. As ρ(G) ≥ x D(G)x and ρ(G ) = x D(G )x, one has ρ(G)−ρ(G ) ≥ x (D(G) − D(G ))x. So, by the claim, and thus ρ(G) > ρ(G ).
In the following, we give some consequences of Theorem 3.2.

Corollary 3.3 ([14]
). Let G be a connected graph with a cut edge uv that is not a pendant edge. Let G be the graph obtained from G by moving all edges at v except uv from v to u. Then ρ(G) > ρ(G ).
So the result follows from Theorem 3.2.
A chain in a graph G is a cycle C such that G − E(C) has exactly |V (C)| components. Length of the cycle C is the length of the chain. The following is Lemma 3.3 of [3].
Corollary 3.4. Let G be a connected graph with a chain C of even length. Let uv be an edge on the chain C.
If z lies on C, this is evident as w = z, otherwise, w is the vertex on C such that its distance to z is minimum among all the distances between vertices of C and z in G. As C is a chain, the shortest path connecting u (v, respectively) and w contains only vertices on C. As the length of C is even, there is a shortest path connecting u and w passing through v or a shortest path connecting v and w passing through u, implying that . So the result follows from Theorem 3.2.
Corollary 3.5. Let H be a graph consisting of two nontrivial connected graphs H 1 and H 2 sharing a unique vertex u such that For any vertex w ∈ V (H 2 ) \ {u}, if all the paths from u to w with the length d H (u, w) pass only through vertices in N 1 or pass only through vertices in N 2 , then ρ(G) > ρ(H).
Note that any shortest path from u to z in H 2 goes through only vertices in N 1 or N 2 . Correspondingly, any shortest path from u to z in G 2 goes through only vertices in If k ≥ 2, then Corollary 3.5 becomes Theorem 2.4 of [16]. Corollary 3.3 may be generalized as the following version.
Theorem 3.6. Let G be the graph obtained from vertex disjoint nontrivial connected graphs G 1 and G 2 with We claim that x vt+1−i − x vi and Γ have common sign for i = 1, . . . , p by induction on i. If i = p, then it follows from (3.1). Suppose that 1 ≤ i ≤ p − 1, and x vt+1−j − x vj and Γ have common sign for i + 1 ≤ j ≤ p. So p j=i+1 (x vt+1−j − x vj ) and Γ have common sign. Thus, from (3.2) ) and Γ have common sign. This, together with the induction assumption that x v t+1−(i+1) − x vi+1 and Γ have common sign, implies that x vt+1−i − x vi and Γ have common sign.
Note that This requires the above common sign to be +. Again, from (3.1) and (3.2), we have . . , p. It follows that for i = 2, . . . , p, As we pass from G to G , the distance between a vertex of V (G 2 ) \ {v t } and a vertex of V (G 1 ) \ {v 1 } is decreased by t − 1, the distance between a vertex of V (G 2 ) \ {v t } and v i for i = 1, . . . , t is decreased by t − 2i + 1, and the distances between all other vertex pairs remain unchanged.
Let G * be the graph obtained from G by moving all the edges at By the similar arguments as above, we have is an isomorphism from G to G * . Let M be the permutation matrix associated to this isomorphism. That is,
Now we present the third graft transformation and consider its effect on the distance spectral radius.
As we pass from G to G , for i = 1, 2, the distance between a vertex of S i and a vertex of V (G 1 ) is decreased by 1, the distance between a vertex of S i and v i is increased by 1, and the distance between any other vertex pair is decreased or remains unchanged. So and thus ρ(G) > ρ(G ).
The following is Lemma 3.4 of [3].
Corollary 3.8. Let G be a connected graph with a chain C of odd length , where ≥ 5. Let uv 1 and uv 2 be two edges on the chain C.
Proof. Let G 1 be the component of G − E(C) containing u, and G 2 be the subgraph of G induced by V (G) \ (V (G 1 ) \ {u}). Let w ∈ V (G 2 ) \ {u}. Denote by z the vertex on C such that its distance to w is minimum among all vertices of C. It is evident that z = w if w lies on C.
As C is a chain, the shortest path connecting v 1 (v 2 , respectively) and z contains only vertices on C. Let P (Q, respectively) be the shortest path connecting v 1 (v 2 , respectively) and z in G. If P and Q are edge disjoint, then . So the result follows from Theorem 3.7.
Corollary 3.9. Let H be a graph consisting of two nontrivial connected graphs H 1 and H 2 sharing a unique vertex u such that E(H) = E(H 1 ) ∪ E(H 2 ). Suppose that uv 1 , . . . , uv k are pendant edges in H 1 , where k ≥ 2 and that N H2 For any vertex w ∈ V (H 2 ) \ {u}, if all the paths from u to w with length d G (u, w) pass only through N 1 or pass only through N 2 , then ρ(G) > ρ(H).
Note that any shortest path from u to z in H 2 goes through only vertices in N 1 or N 2 . Correspondingly, any shortest path from u to z in G 2 goes through only vertices in N 1 or N 2 . Consequently, d G (v k−1 , z) = d G (v k , z). Now by Theorem 3.7, ρ(G) > ρ(H).
We remark that Corollary 3.9 and Theorem 3.7 are equivalent. If k ≥ 3, then Corollary 3.9 is just Theorem 2.3 of [16].

Graphs minimizing the distance spectral radius
First we determine the graphs that minimize the distance spectral radius among all homeomorphically irreducible unicylic graphs on n ≥ 6 vertices. Lemma 4.1 ([9]). For k ≥ 2 and 1 ≤ a 1 ≤ a 2 − 2, let G be a graph obtained from a connected graph G 0 with two vertices u 1 and u 2 such that N G0 (u 1 ) \ {u 2 } ⊆ N G0 (u 2 ) \ {u 1 }, by attaching a i pendant vertices to u i for each i = 1, 2. Let G be the graph obtained from G by moving one pendant edge at u 2 from u 2 to u 1 . Then ρ(G) < ρ(G ).
Let U n be a unicylic graph on n vertices obtained from a triangle K 3 with V (K 3 ) = {v 1 , v 2 , v 3 }, by attaching a pendant vertex to v i for i = 1, 2, respectively, and attaching n − 5 pendant vertices to v 3 . Proof. Let G be a homeomorphically irreducible unicylic graph on n vertices that minimizes the distance spectral radius.
Let g be the girth of the unique cycle C of G. Let u be a vertex on C. Since G is a homeomorphically irreducible unicylic graph, we have deg G (u) ≥ 3. Let v 1 , v 2 be two neighbors of u on C.
Suppose that g ≥ 4. Suppose that g is even. Let G be the graph obtained from G by moving all the edges at v 1 except uv 1 from v 1 to u. Note that G is a homeomorphically irreducible unicylic graph on n vertices. By Theorem 3.2 or Corollary 3.4, ρ(G ) < ρ(G), a contradiction. Thus g is odd. Let G be the graph obtained from G by moving all the edges at v i except uv i from v i to u for each i = 1, 2. Obviously, G is a homeomorphically irreducible unicylic graph on n vertices. By Theorem 3.7 or Corollary 3.8, we have ρ(G ) < ρ(G), also a contradiction. It thus follows that g = 3.
Suppose that G has an edge, say vw, outside C that is not a pendant edge. Evidently, vw is a cut edge of G. Let G * be the graph obtained from G by moving all the edges at w except vw from w to v. It is obvious that G * is a homeomorphically irreducible unicylic graph on n vertices. By Theorem 3.2 or Corollary 3.3, ρ(G * ) < ρ(G), a contradiction. Thus, every edge of G outside C is a pendant edge. That is, G is a unicylic graph obtainable from a triangle K 3 with V (K 3 ) = {v 1 , v 2 , v 3 } by attaching a i pendant vertices to v i for i = 1, 2, 3, where 1 ≤ a 1 ≤ a 2 ≤ a 3 .
If n = 6, 7, then G ∼ = U n . Suppose that n ≥ 8 and a 2 ≥ 2. Let G be the graph obtained from G by moving one pendant edge at v 2 from v 2 to v 3 . Obviously, G is a homeomorphically irreducible unicylic graph on n vertices. By Lemma 4.1, ρ( G) < ρ(G), a contradiction. So a 2 = 1. That is, a 1 = a 2 = 1 and a 3 = n − 5, i.e., G ∼ = U n .
In the following, we determine the trees that minimize the distance spectral radius among all trees on n vertices with given number of vertices of degree two.
Let G be a connected graph with v ∈ V (G). For k, ≥ 0, let G(v, k, ) be the graph obtained from G by attaching two paths P k and P at one end vertices to v. The following lemma was established in [13], for which a simple argument was given in [15]. Lemma 4.3. Let G be a connected graph with v ∈ V (G). If k ≥ ≥ 1, then ρ(G(v, k, )) < ρ(G(v, k + 1, − 1)).
A tree is called starlike if it has exactly one vertex of degree at least three; this vertex is called the branching vertex. If the branch vertex has degree s, we call it an s-starlike tree. For an s-starlike tree T on n vertices with branching vertex u, each path connecting u and a pendant vertex is called a leg. Denote by a 1 , . . . , a s the lengths of the s legs of T . Then a 1 + · · · + a s = |E(T )| = n − 1. Assume that a 1 ≥ · · · ≥ a s . If a 1 − a s = 0, 1, then the multiset {a 1 , . . . , a s } composes of n−1 s + 1 with multiplicity r and n−1 s with multiplicity s − r, where r = n − 1 − s n−1 s . In this case, we call it an s-starlike tree of almost equal leg lengths, denoted by S n,s .
Let T be a tree on n vertices with t vertices of degree two. If t = n − 2, then T ∼ = P n . Note that t = n − 3 is impossible, since T has at least two pendant vertices, and the remaining unique vertex has degree 2(n − 1) − 2(n − 3) − 1 · 2 = 2.
Theorem 4.4. Let T be a tree on n vertices with t vertices of degree two, where 0 ≤ t ≤ n − 4. Then ρ(T ) ≥ ρ(S n,n−t−1 ) with equality if and only if T ∼ = S n,n−t−1 .
Proof. Let T be a tree on n vertices with t vertices of degree two that minimizes the distance spectral radius. Since 0 ≤ t ≤ n − 4, the maximum degree of T is at least three.
Suppose that there are at least two vertices of degree at least three in T . Then we choose two such vertices, say u and v, by requiring that the distance between them is as small as possible. Let P be the path connecting u and v. If u and v are not adjacent, then each vertex on P except u and v has degree two. Let w be the vertex adjacent to v on P (w = u if u and v are adjacent). Let T be the tree obtained from T by moving all the edges at v except wv from v to u. It is easily seen that T possesses t vertices of degree two. By Theorem 3.6, ρ(T ) < ρ(T ), a contradiction. Thus, T has exactly one vertex of degree at least three. That is, T is an s-starlike tree for some s. Assume a 1 , . . . , a s are the lengths of the legs, where a 1 ≥ · · · ≥ a s . Then s i=1 a i = n − 1 and s i=1 (a i − 1) = t. So s = n − t − 1. By Lemma 4.3, a 1 − a n−t−1 = 0, 1. That is, T is an (n − t − 1)-starlike tree of almost equal leg lengths, or T ∼ = S n,n−t−1 .