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The normalized distance Laplacian of a graph G is defined as D L (G) = T (G) − 1 / 2 (T (G) − D (G)) T (G) − 1 / 2 where D (G) is the matrix with pairwise distances between vertices and T (G) is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius. [ABSTRACT FROM AUTHOR]

Abstract In a network or a graph, the total communicability (TC) 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. [ABSTRACT FROM AUTHOR]

Abstract In this note, with a purely geometric approach, the covering radius of the group PGL (3 , q) is determined. Also, a new proof establishing the covering radii of PGL (2 , q) and AGL (1 , q) is provided. [ABSTRACT FROM AUTHOR]

Abstract A permutative matrix is a square matrix such that every row is a permutation of the first row. A circulant matrix is a matrix where each row is a cyclic shift of the row above to the right. The Guo's index λ 0 of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with λ 0 is realizable. The Guo's index of some permutative matrices is obtained. Our results are constructive. Some examples designed using MATLAB are given at the end of the paper. [ABSTRACT FROM AUTHOR]

Abstract Let λ (D) be the A α spectral radius of digraph D , and let G n r be the set of digraphs with order n and dichromatic number r. In this paper, we characterize the digraph which has the maximal A α spectral radius in G n r. Moreover, we also determine the unique digraph which attains the maximum (resp. minimum) A α spectral radius among all strongly connected bicyclic digraphs. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For any real α ∈ [ 0 , 1 ] , Nikiforov (2017) [7] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let u and v be two vertices of a connected graph G . Suppose that u and v are connected by a path w 0 ( = v ) w 1 ⋯ w s − 1 w s ( = u ) where d ( w i ) = 2 for 1 ≤ i ≤ s − 1 . Let G p , s , q ( u , v ) be the graph obtained by attaching the paths P p to u and P q to v . Let s = 0 , 1 . Nikiforov and Rojo (2018) [9] conjectured that ρ α ( G p , s , q ( u , v ) ) < ρ α ( G p − 1 , s , q + 1 ( u , v ) ) if p ≥ q + 2 . In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal A α -spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal A α -spectral radius with fixed order and matching number. These results generalize some known results. [ABSTRACT FROM AUTHOR]

Let t ≥ 3 and G be a graph of order n , with no K 2 , t minor. If n > 400 t 6 , then the spectral radius μ ( G ) satisfies μ ( G ) ≤ t − 1 2 + n + t 2 − 2 t − 3 4 , with equality if and only if n ≡ 1 ( mod t ) and G = K 1 ∨ ⌊ n / t ⌋ K t . For t = 3 the maximum μ ( G ) is found exactly for any n > 40000 . [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n and D ( G ) be its distance matrix. In this paper, we characterize the unique graphs whose distance spectral radius attains the maximum and minimum among all complements of trees. Furthermore, we determine the unique graphs whose least distance eigenvalues attains the maximum and minimum among all complements of trees. [ABSTRACT FROM AUTHOR]

A bag B a g p , q is a graph obtained from a complete graph K p by replacing an edge uv by a path P q . In this paper, we show that for all the connected graphs of order n ≥ 5 with signless Laplacian index q 1 ( G ) and radius r a d ( G ) , q 1 ( G ) ⋅ r a d ( G ) is maximum for and only for the graph B a g n − 2 s + 3 , 2 s − 1 , where s = ⌈ n / 4 ⌉ . This solves a conjecture in [6] on the signless Laplacian index involving the radius. [ABSTRACT FROM AUTHOR]

Let ρ ( D ( G )) denote the distance spectral radius of a graph G and ∂ ( G → ) denote the distance signless Laplacian spectral radius of a digraph G → . Let G n , k D be the set of all k -connected graphs of order n with diameter D . In this paper, we first determine the unique graph with minimum distance spectral radius in G n , k D ; we then give sharp upper and lower bounds for the distance signless Laplacian spectral radius of strongly connected digraphs; we also characterize the digraphs having the maximal and minimal distance signless Laplacian spectral radii among all strongly connected digraphs; furthermore, we determine the extremal digraph with the minimal distance signless Laplacian spectral radius with given dichromatic number. [ABSTRACT FROM AUTHOR]

Recently, Niu et al. (2015) [10] determined the extremal graphs with minimum distance Laplacian spectral radius among n -vertex bipartite graphs with given matching number. However, a more natural problem is left open: Among all n -vertex graphs with a given matching number m , how about the lower bound of their distance Laplacian spectral radii and which graphs minimize the distance Laplacian spectral radii? In this article, we solve this problem completely. Denote by K m ∨ K n − m ‾ the graph obtained from a complete graph K m and n − m isolated vertices by adding m ( n − m ) edges joining each isolated vertex to all vertices of K m . Let G be a connected graph of order n and matching number m , ρ L ( G ) the distance Laplacian spectral radius of G . In this paper, we prove that if m = ⌊ n 2 ⌋ , then ρ L ( G ) ≥ n , with equality if and only if G = K n ; and if 1 ≤ m ≤ ⌊ n 2 ⌋ − 1 , then ρ L ( G ) ≥ 2 n − m , with equality if and only if G ∈ Γ , where Γ denotes the set of n -vertex graphs consisting of K m ∨ K n − m ‾ together with all possible graphs obtained from K m ∨ K n − m ‾ by deleting some edges of K m . [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient condition on the spectral radius for a graph to be Hamiltonian. [ABSTRACT FROM AUTHOR]

In the paper, we will determine the graphs with maximal spectral radius among all the trees with n vertices and independence number α for ⌈ n 2 ⌉ ≤ α ≤ n − 1 . [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n ≥ 3 . The remoteness ρ = ρ ( G ) is the maximum, over all vertices, of the average distance from a vertex to all others. The radius r = r ( G ) is the minimum, over all vertices, of the eccentricity of a vertex. Aouchiche and Hansen (2011) conjectured that ρ − r ≥ 3 − n 4 if n is odd and ρ − r ≥ 2 n − n 2 4 ( n − 1 ) if n is even. In this paper, we confirm this conjecture. In addition, we completely characterize extremal graphs attaining the lower bound. [ABSTRACT FROM AUTHOR]

Let G be a simple, undirected graph with vertex set V . For v ∈ V and r ≥ 1 , we denote by B G , r ( v ) the ball of radius r and centre v . A set C ⊆ V is said to be an r - identifying code in G if the sets B G , r ( v ) ∩ C , v ∈ V , are all nonempty and distinct. A graph G which admits an r -identifying code is called r -twin-free or r - identifiable , and in this case the smallest size of an r -identifying code in G is denoted by γ r I D ( G ) . We study the number of different optimal r -identifying codes C , i.e., such that | C | = γ r I D ( G ) , that a graph G can admit, and try to construct graphs having “many” such codes. [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple connected weighted graph on n vertices, in which the edge weights are positive definite matrices. The eigenvalues of G are the eigenvalues of its adjacency matrix. In this note, we present a correction in equality part in Theorem 2 [S. Sorgun, S. Büyükköse, The new upper bounds on the spectral radius of weighted graphs, Appl. Math. Comput. 218 (2012) 5231–5238]. In addition, some related results are also provided. [ABSTRACT FROM AUTHOR]

Abstract: For , and , let be the minimum value satisfying that for any connected graph of order with radius ; if no such graph exists, we set . For and , let . In this paper, we investigate the behavior of the function and determine some exact values of when or is small. [Copyright &y& Elsevier]

Abstract: We give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible. Then we apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. Some known results in the literature are generalized and improved. [Copyright &y& Elsevier]

Abstract: Let be a nonnegative irreducible matrix with diagonal entries 0. The largest eigenvalue of M is called the spectral radius of the matrix M, denoted by . In this paper, we give two sharp upper bounds of the spectral radius of matrix M. As corollaries, we give two sharp upper bounds of the distance matrix of a graph. [Copyright &y& Elsevier]

Abstract: The signless Laplacian spectral radius of a graph G is the largest eigenvalue of its signless Laplacian matrix. In this paper, the first four smallest values of the signless Laplacian spectral radius among all connected graphs with maximum clique of size greater than or equal to 2 are obtained. [Copyright &y& Elsevier]

Abstract: Let G be a simple connected graph of order n with degree sequence in non-increasing order. The signless Laplacian spectral radius of G is the largest eigenvalue of its signless Laplacian matrix . In this paper, we give a sharp upper bound on the signless Laplacian spectral radius in terms of , which improves and generalizes some known results. [Copyright &y& Elsevier]

Abstract: The distance spectral radius of a graph G is the largest eigenvalue of the distance matrix . In this paper, we characterize the graph with minimum distance spectral radius among trees with fixed number of pendent vertices. [Copyright &y& Elsevier]

Abstract: The spectral radius of a graph G is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph G with is either a dagger, an open quipu, or a closed quipu. The reverse statement is not true. Many open quipus and closed quipus have spectral radii greater than . In this paper we proved the following results. For any open quipu G on n vertices () with spectral radius less than , its diameter satisfies . This bound is tight. For any closed quipu G on n vertices () with spectral radius less than , its diameter satisfies . The upper bound is tight while the lower bound is asymptotically tight. Let be a graph with minimal spectral radius among all connected graphs on n vertices with diameter D. For and , we proved that is the unique graph obtained by attaching two paths of lengths and to a pair of antipodal vertices of the even cycle . This result is tight. Thus we settled a conjecture of Cioabaˇ–van Dam–Koolen–Lee, who previously proved a special case for and n large enough. [Copyright &y& Elsevier]

Abstract: In this article, we determine the unique graph with maximal distance spectral radius in the class of graphs without a pendent vertex. [Copyright &y& Elsevier]

Abstract: In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [8]. [Copyright &y& Elsevier]

Abstract: The power of a graph , denoted by , is the graph with the same vertex set as , such that two vertices are adjacent in if and only if their distance is at most in . In this paper, we give bounds on the first two largest Laplacian eigenvalues of the second power of a general graph, and on the second power of a tree. We also give a Nordhaus–Gaddum-type inequality for the Laplacian spectral radius of . [Copyright &y& Elsevier]

Abstract: A graph is said to have a small spectral radius if it does not exceed the corresponding Hoffmann limit value. In the case of (signless) Laplacian matrix, the Hoffmann limit value is equal to , with ϵ being the real root of . Here the spectral characterization of connected graphs with small (signless) Laplacian spectral radius is considered. It is shown that all connected graphs with small Laplacian spectral radius are determined by their Laplacian spectra, and all but one of connected graphs with small signless Laplacian spectral radius are determined by their signless Laplacian spectra. [Copyright &y& Elsevier]

Abstract: The Laplacian spectral radius of a graph G is the largest eigenvalue of its Laplacian matrix. In this paper, the first three smallest values of the Laplacian spectral radii among all connected graphs with clique number are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, we determine the graph with the largest spectral radius among all the tricyclic graphs with a maximum matching of cardinality . [Copyright &y& Elsevier]

Abstract: The spectra of weighted graphs are given attention by some authors because the graphs in the design of networks and electronic circuits are usually weighted. In this short paper, we completely determine the spectra of weighted double stars. We also give the weighted double star that achieves the maximal spectral radius. [Copyright &y& Elsevier]

Abstract: Let denote the distance matrix of a connected graph with order , where is equal to the distance between and in . The largest eigenvalue of is called the distance spectral radius of graph , denoted by . In this paper, some graft transformations that decrease or increase are given. With them, for the graphs with both order and pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when ; for , the extremal graphs with the maximum distance spectral radius are completely characterized. [Copyright &y& Elsevier]

Abstract: If G is a connected undirected simple graph on n vertices and edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if . Let be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree . Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when , and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices. [Copyright &y& Elsevier]

Abstract: Let be a simple, connected and undirected graph with vertex set and edge set . Also let be the distance matrix of a graph (Janežič et al., 2007) . Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph. A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009)  and Papendieck and Recht (2000) . Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002)  and Das (2009) , respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. [Copyright &y& Elsevier]

Abstract: When a graph is drawn in a classical manner, its vertices are shown as small disks and its edges with a positive width; zero-width edges and zero-size vertices exist only in theory. Let r denote the radius of the disks that show vertices and w the width of edges. We give a list of conditions that make such a drawing good and that apply to not necessarily planar graphs. We show that if , a vertex must have constant degree for a drawing to satisfy the conditions, and if , a vertex can have any degree. We also give an algorithm that, for a given drawing and values for r and w, determines whether the bold drawing satisfies the conditions. Furthermore, we show how to maximize r and/or w without violating the conditions in polynomial time. [Copyright &y& Elsevier]

Abstract: Let G be a simple graph with vertices , of degrees , respectively. Let A be the -adjacency matrix of G and D be the diagonal matrix diag. is called the signless Laplacian of G. The largest eigenvalue of is called the signless Laplacian spectral radius or Q-spectral radius of G. Denote by the chromatic number for a graph G. In this paper, for graphs with order n, the extremal graphs with both the given chromatic number and the maximal Q-spectral radius are characterized, the extremal graphs with both the given chromatic number and the minimal Q-spectral radius are characterized as well. [Copyright &y& Elsevier]

Abstract: Let be a digraph with vertex set . A partition of into acyclic sets is called a -coloring of . The minimum integer for which there exists a -coloring of is the dichromatic number of the digraph . Denote the set of the digraphs of order with the dichromatic number . In this note, we characterize the digraph which has the maximal spectral radius in . Our result generalizes the result of by Feng et al. [Copyright &y& Elsevier]

Abstract: Vizing established an upper bound on the size of a graph of given order and radius. We find a sharp upper bound on the size of a bipartite graph of given order and radius. [Copyright &y& Elsevier]

Abstract: In this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices. [Copyright &y& Elsevier]

Abstract: Let be a strong digraph. The strong distance between two vertices and in , denoted by , is the minimum size (the number of arcs) of a strong subdigraph of containing and . For a vertex of , the strong eccentricity is the strong distance between and a vertex farthest from . The minimum strong eccentricity among all vertices of is the strong radius, denoted by , and the maximum strong eccentricity is the strong diameter, denoted by . The optimal strong radius (resp. strong diameter) (resp. ) of a graph is the minimum strong radius (resp. strong diameter) over all strong orientations of . Juan et al. (2008) [Justie Su-Tzu Juan, Chun-Ming Huang, I-Fan Sun, The strong distance problem on the Cartesian product of graphs, Inform. Process. Lett. 107 (2008) 45–51] provided an upper and a lower bound for the optimal strong radius (resp. strong diameter) of the Cartesian products of any two connected graphs. In this work, we determine the exact value of the optimal strong radius of the Cartesian products of two connected graphs and a new upper bound for the optimal strong diameter. Furthermore, these results are also generalized to the Cartesian products of any () connected graphs. [Copyright &y& Elsevier]

Abstract: In this paper, we characterize all extremal connected bicyclic graphs with the largest signless Laplacian spectral radius in the set of all connected bicyclic graphs with prescribed degree sequences. Moreover, the signless Laplacian majorization theorem is proved to be true for connected bicyclic graphs. As corollaries, all extremal connected bicyclic graphs having the largest signless Laplacian spectral radius are obtained in the set of all connected bicyclic graphs of order (resp. all connected bicyclic graphs with a specified number of pendant vertices, and all connected bicyclic graphs with given maximum degree). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we give some results on Laplacian spectral radius of graphs with cut vertices, and as their applications, we also determine the unique graph with the largest Laplacian spectral radius among all unicyclic graphs with vertices and diameter , . [ABSTRACT FROM AUTHOR]

Abstract: A connected graph is called a quasi-k-cyclic graph, if there exists a vertex such that is a k-cyclic graph (connected with cyclomatic number k). In this paper we identify in the set of quasi-k-cyclic graphs (for ) those graphs whose spectral radius of the adjacency matrix (and the signless Laplacian if ) is the largest. In addition, for quasi-unicyclic graphs we identify as well those graphs whose spectral radius of the adjacency matrix is the second largest. [Copyright &y& Elsevier]

Abstract: The independence number of is defined as the maximum cardinality of a set of pairwise non-adjacent vertices which is called an independent set. In this paper, we characterize the graphs which have the minimum spectral radius among all the connected graphs of order with independence number . [Copyright &y& Elsevier]

Abstract: Restricted to the bicyclic graphs with prescribed degree sequences, we determine the (unique) graph with the largest spectral radius with respect to the adjacency matrix. [Copyright &y& Elsevier]

Abstract: In this paper, we show that among all the connected graphs with n vertices and k cut vertices, the maximal signless Laplacian spectral radius is attained uniquely at the graph , where is obtained from the complete graph by attaching paths of almost equal lengths to all vertices of . We also give a new proof of the analogous result for the spectral radius of the connected graphs with n vertices and k cut vertices (see [A. Berman, X.-D. Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B 83 (2001) 233–240]). Finally, we discuss the limit point of the maximal signless Laplacian spectral radius. [Copyright &y& Elsevier]

Abstract: In this paper, we give a complete characterization of the extremal graphs with maximal Laplacian spectral radius among all unicyclic graphs with given order and given number of pendent vertices. Then we study the Laplacian spectral radius of unicyclic graphs with given independence number and characterize the extremal graphs completely. [Copyright &y& Elsevier]

Abstract: A semiregular tree is a tree where all non-pendant vertices have the same degree. Among all semiregular trees with fixed order and degree, a graph with minimal (adjacency/Laplacian) spectral radius is a caterpillar. Counter examples show that the result cannot be generalized to the class of trees with a given (non-constant) degree sequence. [Copyright &y& Elsevier]

Abstract: We consider the set of unicyclic graphs with prescribed degree sequence. In this set we determine the (unique) graph with the largest spectral radius (or index) with respect to the adjacency matrix. In addition, we give a conjecture about the (unique) graph with the largest index in the set of connected graphs with prescribed degree sequence. [Copyright &y& Elsevier]

Abstract: Let be a graph with vertices and be the largest eigenvalue of the adjacency matrix of . We study how large can be when does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of open problems. [Copyright &y& Elsevier]

Abstract: We study graphs with spectral radius at most and refine results by Woo and Neumaier [R. Woo, A. Neumaier, On graphs whose spectral radius is bounded by , Graphs Combinatorics 23 (2007) 713–726]. We study the limit points of the spectral radii of certain families of graphs, and apply the results to the problem of minimizing the spectral radius among the graphs with a given number of vertices and diameter. In particular, we consider the cases when the diameter is about half the number of vertices, and when the diameter is near the number of vertices. We prove certain instances of a conjecture posed by Van Dam and Kooij [E. R. Van Dam, R. E. Kooij, The minimal spectral radius of graphs with a given diameter, Linear Algebra Appl. 423 (2007) 408–419] and show that the conjecture is false for the other instances. [Copyright &y& Elsevier]