aa r X i v : . [ m a t h . C O ] F e b MAXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANARGRAPHS AYSEL EREY
Abstract.
We show that the kite graph K ( n )4 uniquely maximizes the distance spectral radiusamong all connected -chromatic planar graphs on n vertices. Introduction
In this article all graphs are finite, simple and undirected. The distance between two vertices u and v of a connected graph G , denoted by d G ( u, v ) , is the size of a shortest path between u and v in G . Let v , . . . , v n be the vertices of a connected graph G . The distance matrix D ( G ) of G is an n by n symmetric matrix given by ( D ( G )) ij = d G ( v i , v j ) . For the rest, we shall assume that all mentioned graphs are connected whenever the distancematrices are concerned. Since D ( G ) is a symmetric matrix, all of its eigenvalues are real. By thePerron-Frobenius Theorem, the largest eigenvalue of D ( G ) is positive and it has multiplicity one.The distance spectral radius ρ ( G ) of G is the largest eigenvalue of D ( G ) . One of the central problemsin the recent study of distance matrices is finding extremal graphs maximizing or minimizing thedistance spectral radius in a given family of graphs, see [2] for a recent survey on the subject. One ofthe first results in this direction is due to [15] where it was shown that the path graph P n uniquelymaximizes the distance spectral radius in the family of graphs on n vertices. In [5], the authorsdetermine the extremal values of the distance spectral radius in the family of cacti with n verticesand k cycles. Extremal values of ρ ( G ) were also determined in [17] when G is a graph on n verticeswith k pendant vertices. Maximal distance spectral radius of a graph was determined in variousclasses of trees such as trees with a fixed maximum degree [16], trees on n vertices and matchingnumber m [12] and trees with given order and number of segments [10]. Minimal distance spectralradius of a graph was determined in graphs on n vertices with k cut vertices (or k cut edges) [19]and in multipartite graphs of order n with t parts [13].While relations between chromaticity of graphs and the spectra of other graph matrices such asLaplacian or adjacency matrices have been extensively studied in the literature, very few results areknown on relations between chromaticity and distance spectra of graphs. Let G k,n denote the familyof connected k -chromatic graphs of order n . To minimize ρ ( G ) of a graph G in G k,n , it suffices toconsider only complete k -partite graphs on n vertices, thanks to Perron-Frobenius Theorem. Usingthis fact, it was shown in [11] that the Turán graph uniquely minimizes distance spectral radiusin G k,n . On the other hand, the problem of maximizing the distance spectral radius of a graph in G k,n is wide open. In this article, we make a contribution to this problem by studying the familyof -chromatic planar graphs which is one of the most interesting and challenging graphs families in Date : February 9, 2021.2010
Mathematics Subject Classification.
Key words and phrases. chromatic number, planar graphs, distance spectral radius. the field of chromatic graph theory. The famous Four Color Theorem which stood as an unsolvedproblem for over a century says that every planar graph is -colorable [3, 14]. v v v v n − v n v n − v n − · · · · · · Figure 1.
The kite graph K ( n )4 .The kite graph K ( n ) k is the graph of order n obtained from a k -clique by attaching a path of length n − k at any vertex of the k -clique (see Fig. 1 for k = 4 ). Kite graphs are known to be extremalgraphs for various graph parameters. For example, in [18] it was shown that K ( n ) k is the uniqueextremal graph with maximum distance spectral radius among graphs with fixed clique number k and order n . Moreover, it is known that K ( n ) k has the largest Wiener index in G k,n and it minimizesthe adjacency spectral radius of a graph in G k,n with k ≥ , see [8] and [7] respectively. It is easyto see that K ( n )3 is the unique graph maximizing the distance spectral radius of a graph in G ,n (seeCorollary 4.2). In this article, we focus on -chromatic planar graphs and our main result is thefollowing: Theorem 1.1.
The kite graph K ( n )4 is the unique graph maximizing the distance spectral radiusamong all connected -chromatic planar graphs of order n .2. Graph Theory Terminology
Let G be a graph with vertex set V ( G ) and edge set E ( G ) . The order of G is | V ( G ) | and its size is | E ( G ) | . We write P n , K n and C n for the path, complete and cycle graphs on n verticesrespectively. The graph K is called the trivial graph . We denote a cycle graph C n by v v · · · v n if E ( C n ) = { v i v i +1 : 1 ≤ i ≤ n − } ∪ { v v n } . A k-clique of a graph G is a subgraph isomorphicto the complete graph K k and the clique number of G is the order of the largest clique in G . Asubset of vertices S ⊆ V ( G ) is called an independent set of G if no two vertices in S are adjacentin G . The degree of a vertex v in G is denoted by deg G ( v ) and the open and closed neighborhoodsof v in G are denoted by N G ( v ) and N G [ v ] respectively. For a subset of vertices S in G , the vertexsubset N G ( S ) consists of all vertices in G which has at least one neighbor in S . Also, we define N G [ S ] = S ∪ N G ( S ) . We say that v is a degree k vertex in G if deg G ( v ) = k . We write ∆( G ) and δ ( G ) for the maximum and minimum degrees of G respectively. An isolated vertex of a graph G isa vertex which has no neighbors in G . A leaf vertex is a vertex of degree and a pendant edge isan edge which contains a leaf vertex. Two vertices u and v in G are called twin vertices if either N G ( u ) = N G ( v ) or N G [ u ] = N G [ v ] . Let G denote the complement of G and rG denote the disjointunion of r copies of G . The join of G and H , denoted by G ∨ H , is the graph obtained from thedisjoint union of G and H by adding all edges uv where u ∈ V ( G ) and v ∈ V ( H ) . We say that agraph G ′ is obtained from G by attaching H to G at v if G ′ = G ∪ H and V ( G ) ∩ V ( H ) = { v } . If H is a subgraph of G , we write G \ H for the subgraph of G induced by V ( G ) \ V ( H ) . A proper k -coloring of a graph G is a function f : V ( G ) → { , . . . , k } such that f ( u ) = f ( v ) for every edge uv ∈ E ( G ) . The chromatic number χ ( G ) of a graph G is the minimum integer k such that G hasa proper k -coloring. A graph G is called k-chromatic if χ ( G ) = k and G is called k-colorable if χ ( G ) ≤ k . We say that G is a k-critical graph if G is k -chromatic and χ ( H ) < χ ( G ) for every proper AXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANAR GRAPHS 3 subgraph H of G . It is well known that if G is k -critical, then δ ( G ) ≥ k − . A graph is called planar if it can be drawn on the plane such that no two edges cross each other. A plane graph is a drawingof a planar graph on the plane such that no two edges cross each other.Let G , . . . , G k be subgraphs of G . We say that G , . . . , G k are vertex (edge) disjoint if no twoof them have a common vertex (edge). We say that G , G , G form three cactus-type cycles in G ifthey are edge disjoint cycles and the edge-induced subgraph of G induced by E ( G ) ∪ E ( G ) ∪ E ( G ) has exactly three cycles in it. A connected graph G is called a cactus if every two cycles in G have atmost one common vertex. Let C ( n, k ) denote the family of cacti of order n having exactly k cycles.3. Preliminaries
The following result is an immediate consequence of the Perron-Frobenius Theorem.
Theorem 3.1.
Let e be an edge of a connected graph G and suppose that G \ e is also connected.Then, ρ ( G \ e ) > ρ ( G ) . Lemma 3.2. [16] Let v be a vertex of a nontrivial graph G . For k, l ≥ , we denote by G ( v, k, l ) the graph obtained from disjoint unions of G , P k and P l by adding edges between v and one of theleaf vertices in both P k and P l . If k ≥ l ≥ , then ρ ( G ( v, k + 1 , l − > ρ ( G ( v, k, l )) . Lemma 3.3. [19] Let u and v be two adjacent vertices of a connected graph G where | E ( G ) | ≥ .For positive integers k and l , let G k,l denote the graph obtained from G by attaching paths of length k at u and length l at v with u and v being leaf vertices of the corresponding paths. If k > l ≥ ,then ρ ( G k,l ) < ρ ( G k +1 ,l − ) ; if k = l ≥ , then ρ ( G k,l ) < ρ ( G k +1 ,l − ) or ρ ( G k,l ) < ρ ( G k − ,l +1 ) .The Saw graph S ( p, q ; l ) is the cactus graph of order p + 2 q + l + 1 and size p + 3 q + l whichis obtained from a path graph on p + q + l edges by replacing p consecutive edges including onependant edge by p triangles and replacing q consecutive edges including the other pendant edge by q triangles, see Fig 2. The following result follows from the proof of Theorem 5.3 in [5]. Theorem 3.4. [5] If G is a graph with maximal distance spectral radius in C ( n, k ) , then G ∼ = S ( p, q ; l ) where p + q = k and l = n − k − . v v v v v v n − v n v v n − v n − · · · · · · v v v v v n − v n − v n v n − v n − v n − · · · · · · Figure 2.
The Saw graphs S (3 , n − (left) and S (2 , n − (right).The Broom tree B ( n )∆ is the tree obtained from a path graph P n − ∆+1 by attaching ∆ − pendantedges at an arbitrary leaf vertex of the path P n − ∆+1 , see Fig 3. In [16], it was shown that for every ∆ > ,(1) ρ ( B ( n )∆ ) < ρ ( B ( n )∆ − ) . Theorem 3.5. [16] Let T be a tree of order n with ∆( T ) = ∆ and T ≇ B ( n )∆ . Then, ρ ( T ) < ρ ( B ( n )∆ ) . AYSEL EREY v v v v n − v n v n − v n − v · · · · · · Figure 3.
The broom tree B ( n )5 .The Perron vector of a graph G is the unique normalized eigenvector with positive coordinateswhich corresponds to the eigenvalue ρ ( G ) . In [11], it was shown that the coordinates of the Perronvector of a graph corresponding to its twin vertices are equal.4. Reduction to critical graphs with attached paths
Lemma 4.1. If G is a graph maximizing the distance spectral radius in G k,n , then G has a k -criticalsubgraph H and G is obtained from H by attaching paths at vertices of H . Moreover, if S is thesubset of vertices of H at which nontrivial paths are attached, then S is an independent set of H . Proof.
It is well known that every k -chromatic graph contains a k -critical subgraph. Let H be a k -critical subgraph of an extremal graph G . If G contains a cycle C having an edge e such that e ∈ E ( C ) \ E ( H ) , then G \ e is a connected, k -chromatic graph on n vertices. By Theorem 3.1, weget ρ ( G \ e ) > ρ ( G ) but the latter contradicts with G being an extremal graph in G k,n . Thus, anextremal graph G must be obtained from a k -critical graph H by attaching trees at some verticesof H . Let T be a tree attached to H at a vertex v . We shall show that T must be a path graph and v is a leaf vertex of T . Suppose on the contrary that either T is not a path graph or deg T ( v ) ≥ .Now, there exist a vertex u of T such that T \ u has at least two components which are paths, say P k and P l where k ≥ l ≥ . Let G ′ = G \ ( V ( P k ) ∪ V ( P l )) , then G = G ′ ( u, k, l ) . By Lemma 3.2, wehave ρ ( G ′ ( u, k + 1 , l − > ρ ( G ′ ( u, k, l )) . Note that G ′ ( u, k + 1 , l − is a graph in G k,n and thelatter inequality contradicts with G being an extremal graph.If H has two adjacent vertices x and y such that two nontrivial paths are attached to H at eachof x and y , then we can apply Lemma 3.3 to obtain another graph in G k,n which has larger distancespectral radius than G . Thus, if S is the subset of vertices of H at which nontrivial paths areattached, then all vertices in S are non-adjacent to each other, that is, S is an independent set of H . (cid:3) Corollary 4.2.
The kite graph K ( n )3 is the unique graph maximizing the distance spectral radiusin the family G ,n . Proof.
Let G be an extremal graph in G ,n . It is well known that -critical graphs are precisely theodd cycles. By Lemma 4.1, the graph G belongs to C ( n, . By Theorem 3.4, ρ ( S (1 , n − ≥ ρ ( G ) with equality if and only if G ∼ = S (1 , n − . Now the result follows since S (1 , n − ∼ = K ( n )3 . (cid:3) -chromatic graphs G with ∆( G ) ≥ Lemma 5.1.
Let n be an integer with n ≥ , then ρ ( B ( n )5 ) < ρ ( K ( n )4 ) . Proof. If n ≤ , then the result follows from direct calculations in Table 1. We may assume that n ≥ . We consider the vertex labellings of K ( n )4 and B ( n )5 in Figures 1 and 3 respectively. Let AXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANAR GRAPHS 5 D = D ( K ( n )4 ) , ρ ( K ( n )4 ) = ρ and D ′ = D ( B ( n )5 ) , ρ ( B ( n )5 ) = ρ ′ . For the Perron vector x of B ( n )5 , wehave x T ( D − D ′ ) x = − x n ( x + x + x n − ) − x n − ( x + x ) − x x + ( x n + x n − + x ) n − X i =3 x i . Note that x = x = x n − = x n , since v , v , v n − , v n are twin vertices of B ( n )5 . So, x T ( D − D ′ ) x = − x + 3 x n − X i =3 x i = 3 x − x + n − X i =3 x i ! . By the eigenequation D ′ x = ρ ′ x , we obtain ρ ′ x = 6 x + n − X j =3 ( j − x j and ρ ′ x i = 4( i − x + n − X j =3 | i − j | x j for ≤ i ≤ n − . Now we find that ρ ′ " − x + n − X i =3 x i = " −
12 + 4 n − X i =3 ( i − x + n − X j =3 " − j −
2) + n − X i =3 | i − j | x j = 2( n − n − x + n − X j =3 (cid:20) n − (cid:18)
32 + j (cid:19) n − j + 8 + j (cid:21) x j . It suffices to check that p j ( n ) = n − (cid:0) + j (cid:1) n − j + 8 + j > for every n ≥ and ≤ j ≤ n − .The roots of the quadratic p j ( n ) are
32 + j + 12 p −
55 + 36 j − j and
32 + j − p −
55 + 36 j − j . Observe that p j ( n ) has non-real roots when j ≥ . . Since p j ( n ) has positive leading coefficient, weobtain that p j ( n ) > for all n ≥ when j ≥ . Now we only need to consider the range ≤ j ≤ .The maximum value of f ( j ) = −
55 + 36 j − j is achieved at the unique critical point of j = 9 / and it is equal to . So, the largest root of p j ( n ) is at most + j + √ . It is easy to see that + j + √ < n because ≤ j ≤ and n ≥ . It follows that p j ( n ) > for all n ≥ and ≤ j ≤ n − . Thus, ρ − ρ ′ ≥ x T ( D − D ′ ) x > . (cid:3) n ρ ( S (3 , n − ρ ( S (2 , n − ρ ( B ( n )5 ) ρ ( K ( n )4 )6 8 .
582 8 . .
830 10 .
830 11 .
828 12 . .
462 15 .
404 16 .
090 17 . .
177 20 .
784 21 .
238 23 . .
808 26 .
940 27 .
206 29 . .
279 33 .
850 33 .
959 36 . .
550 41 .
503 41 .
475 44 . Table 1.
Maximal distance spectral radii of certain saw, broom and kite graphslimited to three decimals.
AYSEL EREY
Lemma 5.2. If G is a connected graph of order n with ∆( G ) = ∆ ≥ , then ρ ( G ) < ρ ( K ( n )4 ) . Proof.
Let u be a vertex of maximum degree and v , . . . , v ∆ be the neighbors of u in G . A minimalconnected spanning subgraph of G containing all edges uv i for i ∈ { , . . . , ∆ } is a spanning tree of G with maximum degree ∆ . For such spanning tree T , we have ρ ( G ) ≤ ρ ( T ) by Theorem 3.1 and ρ ( T ) ≤ ρ ( B ( n )∆ ) by Theorem 3.5. Also, ρ ( B ( n )∆ ) ≤ ρ ( B ( n )5 ) by the inequality in (1) and ρ ( B ( n )5 ) <ρ ( K ( n )4 ) by Lemma 5.1. Thus, we obtain that ρ ( G ) ≤ ρ ( T ) ≤ ρ ( B ( n )∆ ) ≤ ρ ( B ( n )5 ) < ρ ( K ( n )4 ) and theresult follows. (cid:3) -chromatic graphs with three cactus-type cycles Lemma 6.1.
For every n ≥ , ρ ( S (3 , n − < ρ ( K ( n )4 ) . Proof. If ≤ n ≤ , then the result follows from direct calculations in Table 1. We may supposethat n ≥ . Let D and D ′ be the distance matrices of K ( n )4 and S (3 , n − with respect to vertexorderings given in Figures 1 and 2 respectively. Let also x = ( x , . . . , x n ) be the Perron vector of D ′ and ρ ′ = ρ ( S (3 , n − . Note that D − D ′ = . . . −
10 0 0 0 1 . . . . . . . . . . . . ... ... ... ... ... . . . ... ... ... . . . . . . − − . . . − . The vertices v and v n − of S (3 , n − are twin vertices, so x = x n − . Now we find that x T ( D − D ′ ) x = (2 x + x + x ) n − X j =5 x j + 2 x n − x + n − X j =5 x j + x n ( x + x ) . It suffices to check that − x + P n − j =5 x j > , as the Perron vector x has positive coordinates. Bythe eigenequation D ′ x = ρ ′ x , we get ρ ′ x = x + x + 2 x + 3 x + 2 x n + n − X k =5 ( k − x k ρ ′ x j = 2( j − x + ( j − x + x n ) + ( j − x + x ) + n − X k =5 | k − j | x k where ≤ j ≤ n − . Let us write ρ ′ − x + n − X j =5 x j = X k ∈{ ,...,n − ,n } f k ( n ) x k . Note that f ( n ) = − P n − j =5 ( j − , f ( n ) = − P n − j =5 ( j − , f ( n ) = − P n − j =5 ( j − , f ( n ) = − f ( n ) , f n ( n ) = − f ( n ) . It is clear that f k ( n ) > for k ∈ { , , , , n } as n ≥ . Also, for ≤ k ≤ n − , f k ( n ) = − ( k −
2) + n − X j =5 | k − j | = 12 n − (cid:18)
32 + k (cid:19) n + k − k + 13 . AXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANAR GRAPHS 7 Also, f k ( n ) is a quadratic polynomial in n with a positive leading coefficient and the roots of f k ( n ) are r = 32 + k + 12 p − k + 44 k − and r = 32 + k − p − k + 44 k − . Observe that r and r are non-real numbers unless . ≤ k ≤ . , so f k ( n ) > for all n when k ≥ . Now, we may suppose that ≤ k ≤ . In this case it is easy to check that n > r and n > r because n ≥ by the assumption. Therefore, we get f k ( n ) > for all n ≥ when ≤ k ≤ . (cid:3) Lemma 6.2.
For every n ≥ , ρ ( S (2 , n − < ρ ( K ( n )4 ) . Proof. If n ≤ , then the result follows from direct calculations in Table 1. We may assume that n ≥ . Let D and D ′ be the distance matrices of K ( n )4 and S (2 , n − with respect to vertexorderings given in Figures 1 and 2 respectively. Let also x = ( x , . . . , x n ) be the Perron vector of D ′ and ρ ′ = ρ ( S (2 , n − . Note that D − D ′ = . . . −
10 0 0 . . . . . . ... ... ... . . . ... ... ... ... ... . . . . . . . . . . . . − − . . . − . Observe that the pairs v , v n − and v n − , v n − of S (2 , n − are twin vertices, so x = x n − and x n − = x n − . Now, x T ( D − D ′ ) x = x n − x n − + n − X j =1 x j + x n x n − − x + n − X j =3 x j . It suffices to only show that x n − − x + n − X j =3 x j > . By the eigenequation D ′ x = ρ ′ x , ρ ′ x n − = x n − + ( n − x n − + ( n − x n + n − X k =1 ( n − − k ) x k ρ ′ x = ( n − x n − + x n − + 2 x n + n − X k =2 ( k − x k ρ ′ x j = ( n − − j ) x n − + ( j − x n − + ( j − x n + n − X k =1 | j − k | x k where ≤ j ≤ n − . Let us write ρ ′ x n − − x + n − X j =3 x j = n X k =1 f k ( n ) x k . AYSEL EREY
Since ρ ′ > and all x k ’s positive, it suffices to show that f k ( n ) > for every k ∈ { , . . . , n } . It isstraightforward to check that f k ( n ) > when k ∈ { , n − , n − , n − , n } , as n ≥ . Suppose that ≤ k ≤ n − , then f k ( n ) = 3( n − − k ) − k −
1) + n − X j =3 | k − j | = 12 n − (cid:18)
12 + k (cid:19) n − k + 2 + k . The roots of the quadratic f k ( n ) are r = 12 + k + 12 p −
15 + 36 k − k and r = 12 + k − p −
15 + 36 k − k . Observe that f k ( n ) has no real roots when k ≥ because −
15 + 36 k − k < for k ≥ . Also,when ≤ k ≤ , it is straightforward to verify that n > r and n > r , as n ≥ . Thus, f k ( n ) > when n ≥ and k ≥ , and the result follows. (cid:3) Lemma 6.3. If G is a connected graph of order n having three cactus-type cycles, then ρ ( G ) <ρ ( K ( n )4 ) . Proof.
Let H a minimal spanning subgraph of G containing three cactus-type cycles in G . So, H is a cactus subgraph of G which belongs to C ( n, . By Theorem 3.1, we have ρ ( H ) ≥ ρ ( G ) . ByTheorem 3.4, we have ρ ( H ) ≤ ρ ( S (2 , n − and ρ ( H ) ≤ ρ ( S (3 , n − . Also, ρ ( K ( n )4 ) is largerthan each of ρ ( S (3 , n − and ρ ( S (2 , n − by Lemmas 6.1 and 6.2. Thus, ρ ( G ) ≤ ρ ( H ) <ρ ( K ( n )4 ) and the result follows. (cid:3) We say that a graph G satisfies the property P if either ∆( G ) ≥ or G contains three cactus-typecycles. By Lemmas 5.2 and 6.3 we have the following: Corollary 6.4. If G is a graph of order n satisfying the property P , then ρ ( G ) < ρ ( K ( n )4 ) -critical planar graphs with exactly four triangles The well known Grünbaum-Aksenov Theorem [1, 9] says that every -chromatic planar graphcontains at least four triangles. In [4], a characterization of planar -critical graphs with exactlyfour triangles was given. In this section we will give this characterization and we follow the definitionsgiven in [4]. Let diamond and tailed diamond graphs be as shown in Fig. 4. We say that an edge e is a diamond edge of a graph G if e belongs to exactly two triangles in G . A diamond expansion of a graph G is defined as follows: delete some diamond edge xy from G , and identify x with a leafvertex of the tailed diamond, and identify y with a degree vertex of the tailed diamond graph. G ⋄ G ♦ u vH Figure 4.
Diamond G ⋄ , tailed diamond G ♦ and Havel’s quasi-edge H .Let TW denote the family of graphs that can be obtained from K by diamond expansions.Observe that every edge of K is a diamond edge and a diamond expansion of K using any edgeof K is isomorphic to the Moser spindle graph M shown in Fig.5. The Moser spindle has two AXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANAR GRAPHS 9 diamond edges, namely aa ′ and bb ′ , and end-vertices of each diamond edge are twin vertices. Usingthe symmetry of the graph, it is easy to see that any diamond expansion of M is isomorphic to thegraph T in Fig. 5. Note that T has two diamond edges, namely, αα ′ and ββ ′ . Also, the graph T contains a disjoint union K ∪ C where each triangle K in the disjoint union contains a diamondedge of T (the thick edges of T in Fig. 5 form a K ∪ C ). Therefore, all diamond expansions of T must contain a K ∪ C as well. K eaa ′ b b ′ d cM β β ′ αα ′ T Figure 5. K , the Moser spindle M and the graph T . Havel’s quasi-edge is the graph denoted by H in Fig. 4. Let u and v be two vertices of degree in H . A Havel’s quasi-edge H expansion of a graph G is defined as follows: delete some diamondedge xy from G , and identify x with the vertex u of H and identify y with the vertex v of H . Let TW denote the family of graphs that can be obtained from a graph in TW by a Havel’s quasi-edge H expansion. Let TW denote the family of graphs that can be obtained from a graph in TW by aHavel’s quasi-edge H expansion. It is clear that every graph in TW ∪ TW contains a K becauseevery graph in TW has two non-adjacent diamond edges and H contains a K which containsneither of the vertices u and v . x z ′ y ′ yz x ′ rQ x z ′ y ′ yz x ′ r sQ x z ′ y ′ yz x ′ Q Figure 6.
Several quadrangulations of the interior of C .Let G be a plane graph. We say that a subgraph P is a patch in G if P is a quadrangulationof the interior of a C shown by xz ′ yx ′ zy ′ , such that all neighbors of x ′ , y ′ and z ′ are inside of P .The patch P is called critical if x ′ , y ′ and z ′ have at least three neighbors and every cycle of lengthfour in P bounds a face. Let v be a vertex of G with N G ( v ) = { x, y, z } . We write G v for the graphobtained from G \ v by inserting a critical patch P whose boundary is a C shown by xz ′ yx ′ zy ′ ,with x ′ , y ′ , z ′ being new vertices, and we say that G v is a critical patch P expansion of G at thevertex v . We refer the reader to [4] for further details of the notions mentioned here. Critical patchexpansions of K using the quadrangulations Q and Q in Fig. 6 is shown in Fig. 7. The graph M ′ in Fig. 7 is also known as the Mycielskian of a triangle . Theorem 7.1. [4] A planar -critical graph has exactly four triangles if and only if it is obtainedfrom a graph in F = TW ∪ TW ∪ TW by replacing several (possibly zero) non-adjacent vertices ofdegree with critical patches. vz yxK x z ′ y ′ yz x ′ rM ′ x z ′ y ′ yz x ′ r sM ′′ Figure 7.
Some critical patch expansions of K at v . ayx b cv zM ay x ′ xy ′ z ′ b czM ay x ′ xy ′ z ′ b czM Figure 8.
Some critical patch expansions of M at v . The graphs M and M contain a K ∪ C shown by bold edges. u u u v v v Figure 9.
The triangular grid graph T ∗ . Lemma 7.2.
Let G be a -critical planar graph. If G contains a triangular grid graph, then either G is the Mycielskian of a triangle M ′ or G satisfies property P . Proof.
Let T ∗ be a triangular grid in G with vertices labelled as in Fig. 9. First, note that V ( G ) \ V ( T ∗ ) = ∅ because the unique -critical graph on six vertices is the wheel graph K ∨ C whichclearly does not contain a T ∗ . Let us suppose that ∆( G ) = 4 . If G \ T ∗ has a cycle C in it, thenthis cycle C and the triangles u v v and v u v are three cactus-type cycles in G . So, for the rest,we may assume that G \ T ∗ is a forest. We consider two cases. Case 1: G \ T ∗ has an isolated vertex v . Since ∆( G ) = 4 and δ ( G ) ≥ , the vertex v has no neighborin { v , v , v } and it must be adjacent to all of u , u , u . Now, T ∗ together with v yields the graph M ′ which is a -critical graph. Note that a -critical graph cannot contain a -critical graph as aproper subgraph. Therefore, G must indeed be isomorphic to M ′ . Case 2: G \ T ∗ has no isolated vertices. Let w and w be two leaves of a connected component of G \ T ∗ . Since ∆( G ) = 4 and δ ( G ) ≥ , the vertices w and w have no neighbors in { v , v , v } and AXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANAR GRAPHS 11 have at least two neighbors in { u , u , u } . Hence, some vertex u i must be adjacent to both of w and w . The edges u i w and u i w together with the unique path joining v to v in G \ T ∗ forma cycle C . Now, the cycle C together with the triangles u v v and v u v form three cactus-typecycles in G . (cid:3) Corollary 7.3.
Every planar -critical graph with exactly four triangles which is different from K , the graph M (the Moser spindle) and the graph M ′ (the Mycielskian of a triangle) satisfies theproperty P . Proof.
As we already observed earlier, every graph in F \ { K , M } contains three vertex disjointcycles. It is easy to see that if a graph contains three vertex disjoint cycles, then so does everycritical patch expansion of the graph. Therefore, we only need to consider critical patch expansionsof K and M . Let G ≇ K be a critical patch expansion of K . Then, G must contain a triangulargrid T ∗ . Moreover, replacing degree vertices of a graph containing a triangular grid T ∗ with criticalpatches yields a graph which also contains triangular grid. Now, by Lemma 7.2, all graphs obtainedfrom a K by such critical patch expansions, except K itself and the graph M ′ , satisfy the property P . Suppose that G is a graph obtained from M by replacing an end-vertex of a diamond edge of M by a critical patch. Let u be the unique vertex of degree four in M . Observe that deg G ( u ) ≥ because u is adjacent to two new vertices in G besides the three end-vertices of the two diamondedges. It is also clear that a critical patch expansion cannot decrease the maximum degree of thegraph. Lastly, let us consider a critical patch expansion G obtained from M by replacing a degree-three vertex which does not belong to a diamond edge. It is straightforward to check that in thiscase G has three vertex disjoint cycles, indeed G contains a K ∪ C , and the result follows. (cid:3) Planar -critical graphs with at least five triangles Lemma 8.1.
Let G be a -critical planar graph. If G has at least five triangles and G contains thefan graph K ∨ P , then G satisfies P . Proof.
Let T , T , T be three triangles in a fan subgraph K ∨ P of G and let V ( T i ) = { u, v i , v i +1 } for i ∈ { , , } . Note that v v , v v / ∈ E ( G ) because otherwise G would contain a K whichcontradicts with G being -critical. We consider two cases. Case 1 : v v ∈ E ( G ) . Let T be the triangle uv v and let T ′ be a triangle different from T , T , T and T . If T ′ does not share an edge with any of the triangles T , T , T and T , then T , T and T ′ are three cactus-type triangles in G . If T ′ shares an edge uv i for some i , then ∆( G ) ≥ deg G ( u ) ≥ and we are done. If T ′ contains the edge v v , then the triangles T , T , T and T ′ form a triangulargrid. By Lemma 7.2, G must satisfy property P because G has at least five triangles and the graph M ′ has exactly four triangles. If T ′ contains the edge v v , then the triangles T, T ′ , T , T form atriangular grid and we are done by Lemma 7.2 again. Similar argument works if T ′ contains theedge v v too. Lastly, suppose that T ′ contains the edge v v . Now, the triangles T, T ′ , T , T forma triangular grid and the result follows from Lemma 7.2. Case 2: v v / ∈ E ( G ) . If there exists a triangle T / ∈ { T , T , T } which does not share an edge withany of T , T or T , then T, T and T are three cactus-type triangles. We may assume that everytriangle G shares an edge with some triangle T i . If there exists a triangle T / ∈ { T , T , T } and T contains some edge uv i or v v , we proceed as in previous case. Let T and T ′ be two distincttriangles different from T , T , T . We may assume that T and T ′ do not contain any of the edges uv i and v v where i ∈ { , , , } . If both of T and T ′ contains the edge v v or v v , then we get deg G ( v ) ≥ or deg G ( v ) ≥ respectively. Without loss of generality, suppose that T contains the edge v v and T ′ contains the edge v v . Now, T, T ′ and T are three cactus-type triangles and weare done. (cid:3) u u v v v u u v v v Figure 10.
Two planar drawings of K ∨ K . Lemma 8.2. If G is a -critical planar graph containing K ∨ K as a subgraph, then G satisfiesproperty P . Proof.
Let u , u , v , v , v be the vertices of a K ∨ K subgraph such that u u ∈ E ( G ) and u i v j ∈ E ( G ) for ≤ i ≤ and ≤ j ≤ . We may assume that N G ( u ) = { u , v , v , v } and N G ( u ) = { u , v , v , v } because otherwise we would have deg G ( u ) ≥ or deg G ( u ) ≥ both ofwhich yields ∆( G ) ≥ . Since G is -critical, G is K -free. So, v i v j / ∈ E ( G ) for every i and j . For i ∈ { , } we recursively define a sequence of vertex subsets as follows: A ( i )1 = N G ( v i ) \ { u , u } and A ( i ) k = N G [ A ( i ) k − ] \ { u , u , v } for k ≥ .Let l i be the largest integer such that A ( i ) l i = A ( i ) l i +1 . Let H i be the subgraph of G induced bythe vertex subset A ( i ) l i ∪ { v } for i ∈ { , } . There are two different planar drawings of K ∨ K (see Fig. 10). The existence of a vertex v such that v ∈ V ( H ) ∩ V ( H ) and v = v violates theplanarity of the graph. Hence, V ( H ) ∩ V ( H ) ⊆ { v } . Note that δ ( G ) ≥ , as G is -critical. Sothe vertex v i has at least one neighbor in V ( H i ) \ { v } and every vertex v ∈ V ( H i ) \ { v i , v } hasat least three neighbors in H i for i ∈ { , } . Now, H i is a subgraph with at least three verticesand deg H i ( v ) ≥ for every vertex v in V ( H i ) \ { v i , v } . The latter shows that H i cannot be aforest and therefore H i must have at least one cycle in it. Let C (1) and C (2) be two cycles in H and H respectively. Let T denote the triangle v u u . Note that V ( C (1) ) ∩ V ( C (2) ) ⊆ { v } , V ( C (1) ) ∩ V ( T ) ⊆ { v } , V ( C (2) ) ∩ V ( T ) = ∅ . Now, the cycles C (1) , C (2) and the triangle T formthree cactus-type cycles. (cid:3) Lemma 8.3. If G is a -critical planar graph with at least five triangles, then G satisfies P . Proof.
If there are no two triangles in G sharing a common edge, then G must have three cactus-typecycles, as G has at least five triangles. Now we may assume that there exist two triangles, say T and T , having a common edge. Let V ( T ) = { v , u, v } and V ( T ) = { v , u, v } . Note that v v / ∈ E ( G ) ,since G is -critical and hence cannot contain a K . We may assume that every triangle T in G different from T and T satisfies the following:(i) T does not share an edge with either of the triangles T and T , and(ii) T does not contain any of the end-vertices of the shared edge uv of T and T because if T contains the edge uv , then G contains a K ∨ K and the results follow from Lemma 8.2;if T contains any of the edges uv , uv , vv , vv , then G contains a K ∨ P and the result followsfrom Lemma 8.1; if T satisfies (i) and contains any of the vertices u or v , then ∆( G ) ≥ . Let T and T be two other triangles of G different from T and T . If T and T are edge-disjoint triangles,then the triangles T , T and T form three cactus-type cycles in G . Now we may assume that the AXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANAR GRAPHS 13 triangles T and T share a common edge. Let T / ∈ { T , T , T , T } be another triangle. Repeatingthe earlier argument, we may assume that T does not share an edge with either of the triangles T and T . Now, T , T and T are three cactus-type cycles. (cid:3) Theorem 8.4.
Let G be a -critical planar graph different from K , M and M ′ . Then, G satisfiesproperty P . Proof.
By Grünbaum-Aksenov Theorem, G has at least four triangles. If G has exactly four triangles,then the result follows from Corollary 7.3. If G has at least five triangles, then the result followsfrom Lemma 8.3. (cid:3) -chromatic graphs containing the Moser spindle e aa ′ bb ′ dcu u t v v k · · · · · · G e aa ′ b ′ b dcu u t v v k · · ·· · · G Figure 11.
Graphs G , G of order n where ≤ k, t ≤ n − and k + t = n − . Lemma 9.1.
For every n ≥ , ρ ( G ) < ρ ( K ( n )4 ) where G is the graph shown in Fig. 11. Proof.
By Lemma 3.3, ρ ( K ( t,k )4 ) ≤ ρ ( K ( n )4 ) , so it suffices to show that show that ρ ( G ) < ρ ( K ( t,k )4 ) .We consider a labelling of the vertices of K ( t,k )4 given in Fig. 12. As we pass from G to K ( t,k )4 , the d a ′ ea b b ′ c v v k u u t · · ·· · · Figure 12.
The graph K ( t,k )4 with a particular labelling of its vertices to be com-pared to G .distance between the vertices d and e decrease by and the distances between all the rest of thevertices increase or stay the same. In particular, the distance between d and b ′ increases by andthe distance between d and c increases by . Hence, for the Perron vector x of G , we get x T (cid:16) D ( K ( t,k )4 ) − D ( G ) (cid:17) x > x d ( x b ′ + 3 x c − x e ) . Note that x b = x b ′ , since b and b ′ are twin vertices of G . By D ( G ) x = ρ ( G ) x , ρ ( G ) ( x b ′ + 3 x c − x e ) = 7( x a + x a ′ ) + 5 x b − x c + 3 x d + 7 x e + t X i =1 (3 i + 7) x u i + k X i =1 (3 i − x v i . To show that the latter is positive, it suffices to check that x b − x c +7 x e > . Using the eigenequationagain, we find that ρ ( G )(5 x b − x c + 7 x e ) is equal to x a + x a ′ ) + 17 x b + 19 x c + 23 x d + 3 x e + t X i =1 (11 i + 15) x u i + k X i =1 (11 i + 19) x v i which is clearly positive. Thus, x T (cid:16) D ( K ( t,k )4 ) − D ( G ) (cid:17) x > and the result follows. (cid:3) Lemma 9.2.
For every n ≥ , ρ ( G ) < ρ ( K ( n )4 ) where G is the graph shown in Fig. 11. Proof.
As in Lemma 9.1, it suffices to show that ρ ( G ) < ρ ( K ( t,k )4 ) . This time we consider thelabelling of the vertices of K ( t,k )4 given in Fig. 13. a ′ d ea b ′ c b v v k u u t · · ·· · · Figure 13.
The graph K ( t,k )4 with a particular labelling of its vertices to be com-pared to G .As we pass from G to K ( t,k )4 , the distance between the vertices d and e decreases by andthe distances between all the other vertices increase or stay the same. In particular, the distancebetween the vertices d and b , and the distance between d and c increase by . Hence, for the Perronvector x of G , we get x T (cid:16) D ( K ( t,k )4 ) − D ( G ) (cid:17) x > x d (2 x b + 2 x c − x e ) . Now it suffices to check that (2 x b + 2 x c − x e ) > . By the eigenequation D ( G ) x = ρ ( G ) x , wecalculate that ρ ( G )(2 x b + 2 x c − x e ) is equal to x a + x a ′ ) + x b + 3 x b ′ + 4 x d + 6 x e + k X i =1 (3 i + 1) x v i + t X i =1 (3 i + 7) x u i which is clearly positive. (cid:3) -chromatic graphs containing the Mycielskian of a triangle We begin with the graph M ′ in Fig. 14 which is obtained from M ′ by attaching paths at threenonadjacent vertices in M ′ . Note that the graph M ′ is isomorphic to M ′ when r = s = t = 1 . Lemma 10.1.
For every n ≥ , ρ ( M ′ ) < ρ ( K ( n )4 ) . AXIMAL DISTANCE SPECTRAL RADIUS OF -CHROMATIC PLANAR GRAPHS 15 v v v r a bdcu u u s w w w t · · · · · ·· · · M ′ v v r a bd cu u s w w t · · ·· · ·· · · K ( r,s,t )4 Figure 14.
The graphs M ′ and K ( r,s,t )4 where r, s, t ≥ and r + s + t = n − . Proof.
Let ρ = ρ ( M ′ ) , ρ ′ = ρ ( K ( r,s,t )4 ) and D = D ( M ′ ) , D ′ = D ( K ( r,s,t )4 ) . By Lemma 3.3, ρ ′ ≤ ρ ( K ( n )4 ) . So, it suffices to show that ρ < ρ ′ . Let us consider a vertex labelling of the graphs as inFig. 14. As we move from M ′ to K ( r,s,t )4 the distance from d to a , b , or c decreases by one; thedistance from d to u , v or w increases by one. Moreover, the distances between all the othervertices increase or stay the same. Therefore, ρ ′ − ρ ≥ x T ( D ′ − D ) x ≥ x d ( x v + x u + x w − x a − x b − x c ) where x is the Perron vector of D . Applying the eigenequation Dx = ρx repetitively, we find that ρ ( x v + x u + x w − x a − x b − x c ) = 2( x a + x b + x c ) − x d ≥ ρ (5( x u + x v + x w ) − x a + x b + x c ) + 12 x d ) ≥ ρ (24( x u + x v + x w ) + 40( x a + x b + x c ) + 3 x d ) and the latter is clearly positive. Thus, ρ ′ > ρ . (cid:3) b vw ac u d d d n − · · · M ′ a cb w v u d d n − · · · K ( n )4 Figure 15.
The graphs M ′ and K ( n )4 . Lemma 10.2.
For every n ≥ , ρ ( M ′ ) < ρ ( K ( n )4 ) . Proof.
Let ρ = ρ ( M ′ ) , ρ ′ = ρ ( K ( n )4 ) and D = D ( M ′ ) , D ′ = D ( K ( n )4 ) . Let us consider a vertexlabelling of the graphs as in Fig. 15. As we move from M ′ to K ( n )4 the distances from w to a and v to u decrease by one, the distance from u to v decreases by one, and the distances between all theother vertices increase or stay the same. In particular, the distance from a to d , u and v increases by , and respectively; the distances from v to b and d increase by one; and the distance from c to u increases by . Therefore, x T ( D ′ − D ) x ≥ x a (2 x d + x v − x w ) + x u (2 x a + 2 x c − x v ) + x v ( x b + x d − x w ) for the Perron vector x of D . By the eigenequation ρx = Dx , we have ρ (2 x d + x v − x w ) = 3 x a + 4 x b + 5 x c + 2 x u + 4 x w + n − X j =1 j − x d j > ,ρ (2 x a + 2 x c − x v ) = x a + 3 x b + 2 x u + 6 x v + 4 x w + n − X j =1 (3 j + 4) x d j > ,ρ ( x b + x d − x w ) = x a + x b + 2 x c + x u + 2 x w + n − X j =1 jx d j > . Thus, ρ ′ − ρ ≥ x T ( D ′ − D ) x > . (cid:3) Proof of Theorem 1.1
We are now ready to prove our main result.
Proof of Theorem 1.1.
Let G be an extremal graph with maximal distance spectral radius amongall connected -chromatic planar graphs of order n . By the proof Lemma 4.1, G is obtained froma -critical planar graph H by attaching paths at vertices of some independent set S of H . If anontrivial path is attached at a degree vertex in S , then ∆( G ) ≥ and Lemma 5.2 shows that G cannot be an extremal graph. So, we may assume that S consists of nonadjacent vertices of degreeat most three. Now we shall consider three cases: Case 1: H is K . The complete graph K has no two non-adjacent vertices. So, G must be K ( n )4 if H is K . Case 2: H is the Moser spindle M . Consider a vertex labelling of M as in Fig. 5. The graph M has eight independent sets consisting of vertices of degree at most three. However, by symmetry, itsuffices to consider the independent sets { a, c } and { a, b } only. In Lemmas 9.1 and 9.2 it was shownthat the distance spectral radius of a graph obtained from M by attaching two paths at the verticesin { a, c } or { a, b } respectively is less than that of K ( n )4 . Therefore, this case is not possible. Case 3: H is the Mycielskian of a triangle M ′ . Using the symmetry and the fact that S does notcontain a degree vertex of M ′ , it suffices to consider the graphs M ′ and M ′ shown in Figures 14and 15. It follows from Lemmas 10.1 and 10.2 that G cannot be an extremal graph in this case. Case 4: H is different from K , M and M ′ . By Theorem 8.4, H satisfies property P and therefore G also satisfies P . By Corollary 6.4, the graph G cannot be an extremal graph.Thus, H must be K and the unique extremal graph is K ( n )4 . (cid:3) References [1] V.A. Aksenov, The extension of a -coloring on planar graphs, Diskret. Analiz
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Department of Mathematics, Gebze Technical University, Kocaeli, Turkey
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