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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 Let G = (V , E) be a simple graph of order n with normalized Laplacian eigenvalues ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 ≥ ρ n = 0. The normalized Laplacian spread of graph G , denoted by ρ 1 − ρ n − 1 , is the difference between the largest and the second smallest normalized Laplacian eigenvalues of graph G. In this paper, we obtain the first four smallest values on ρ 2 of graphs. Moreover, we give a lower bound on ρ 2 of connected bipartite graph G except the complete bipartite graph and characterize graphs for which the bound is attained. Finally, we present some bounds on the normalized Laplacian spread of graphs and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

We study the Laplacian and the signless Laplacian energy of connected unicyclic graphs, obtaining a tight upper bound for this class of graphs. We also find the connected unicyclic graph on n vertices having largest (signless) Laplacian energy for 3 ≤ n ≤ 13 . For n ≥ 11 , we conjecture that the graph consisting of a triangle together with n − 3 balanced distributed pendent vertices is the candidate having the maximum (signless) Laplacian energy among connected unicyclic graphs on n vertices. We prove this conjecture for many classes of graphs, depending on σ , the number of (signless) Laplacian eigenvalues bigger than or equal to 2. [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with maximum degree Δ and minimum degree δ . Let ( d ) = ( d 1 , d 2 , … , d n ) and ( d ⁎ ) = ( d 1 ⁎ , d 2 ⁎ , … , d n ⁎ ) be the sequences of degrees and conjugate degrees of G . We define π = ∑ i = 1 n d i and π ⁎ = ∑ i = 1 n d i ⁎ , and prove that π ⁎ ≤ L E L ≤ I E ≤ π where LEL and IE are, respectively, the Laplacian-energy-like invariant and the incidence energy of G . Moreover, we prove that π − π ⁎ > ( δ / 2 ) ( n − Δ ) for a certain class of graphs. Finally, we compare the energy of G and π , and present an upper bound for the Laplacian energy in terms of degree sequence. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n with diameter d . Remoteness ρ of G is the maximum average distance from a vertex to all others and ∂ 1 ≥ ⋯ ≥ ∂ n are the distance eigenvalues of G . Aouchiche and Hansen (0000), Aouchiche and Hansen conjectured that ρ + ∂ 3 > 0 when d ≥ 3 and ρ + ∂ ⌊ 7 d 8 ⌋ > 0 . In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on ∂ n + ρ and ∂ 1 − ρ when G ≇ K n and the extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

The number of spanning trees of a graphGis the total number of distinct spanning subgraphs ofGthat are trees. In this paper, we present sharp upper bounds for the number of spanning trees of a graph with given matching number. [ABSTRACT FROM AUTHOR]

The inverse degree of graph G is defined as I D ( G ) = ∑ v ∈ V ( G ) 1 d G ( v ) where d G ( v ) is the degree of vertex v in G . In this paper we have determined some upper and lower bounds on the inverse degree I D ( G ) for a connected graph G in terms of other graph parameters, such as chromatic number, clique number, connectivity, number of cut edges, matching number. Also the corresponding extremal graphs have been completely characterized. [ABSTRACT FROM AUTHOR]

The Wiener polarity index W p ( G ) of a graph G is the number of unordered pairs of vertices { u ,  v } in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in [Y. Zhang, Y. Hu, 2016] [38]. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus–Gaddum-type results for W p ( G ). Our lower bound on W p ( G ) + W p ( G ¯ ) is always better than the previous lower bound given by Zhang and Hu. [ABSTRACT FROM AUTHOR]

The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

The Graovac–Ghorbani index ( ABC GG ), was introduced by Graovac and Ghorbani, (2010). In this paper we present some upper and lower bounds on the ABC GG index and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

The zeroth-order Randić index and the sum-connectivity index are very popular topological indices in mathematical chemistry. These two indices are based on vertex degrees of graphs and attracted a lot of attention in recent years. Recently Li and Li (2015) studied these two indices for trees of order n . In this paper we obtain a relation between the zeroth-order Randić index and the sum-connectivity index for graphs. From this we infer an upper bound for the sum-connectivity index of graphs. Moreover, we prove that the zeroth-order Randić index is greater than the sum-connectivity index for trees. Finally, we show that R 2,  α ( G ) is greater or equal R 1,  α ( G ) ( α ≥ 1) for any graph and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Let G be a connected threshold graph of order n with m edges and trace T . In this paper we give a lower bound on Laplacian energy in terms of n, m and T of G . From this we determine the threshold graphs with the first four minimal Laplacian energies. Moreover, we obtain the threshold graphs with the largest and the second largest Laplacian energies. [ABSTRACT FROM AUTHOR]

Let G be a bipartite graph of order n with m edges. The energy E ( G ) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A . In 1974, one of the present authors established lower and upper bounds for E ( G ) in terms of n, m , and det A . Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m , and the first Zagreb index is obtained, and the extremal graphs characterized. [ABSTRACT FROM AUTHOR]

The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G , defined as D Q ( G ) = T r ( G ) + D ( G ) , where D ( G ) is the distance matrix of G and T r ( G ) is the diagonal matrix of vertex transmissions of G . In this paper we determine upper and lower bounds on the minimal and maximal entries of the principal eigenvector of D Q ( G ) and characterize the extremal graphs. In addition, we obtain a lower bound on the distance signless Laplacian spectral radius of G based on its order and independence number, and characterize the extremal graph. [ABSTRACT FROM AUTHOR]

The first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper we present the lower bound on M1 and M2 among all unicyclic graphs of given order, maximum degree, and cycle length, and characterize graphs for which the bound is attained. Moreover, we obtain some relations between the Zagreb indices for unicyclic graphs. [ABSTRACT FROM AUTHOR]

Let S z ( G ) and ξ c ( G ) be the Szeged index and the eccentric connectivity index of a graph G , respectively. In this paper we obtain a lower bound on S z ( T ) − ξ c ( T ) by double counting on some matrix and characterize the extremal graphs. From this result we compare the Szeged index and the eccentricity connectivity index of trees. For bipartite graphs we also compare the Szeged index and the eccentricity connectivity index. Moreover, we show that S z ( G ) − ξ c ( G ) ≥ − 4 for bipartite graphs and this result is not true in the general case. Finally, we classify the bipartite graphs G in which S z ( G ) − ξ c ( G ) ∈ { − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 } . [ABSTRACT FROM AUTHOR]

Let G be a simple graph of order n with vertex set V = { v 1 , v 2 , … , v n } . Also let μ 1 ( G ) ≥ μ 2 ( G ) ≥ ⋯ ≥ μ n − 1 ( G ) ≥ μ n ( G ) = 0 and q 1 ( G ) ≥ q 2 ( G ) ≥ ⋯ ≥ q n ( G ) ≥ 0 be the Laplacian eigenvalues and signless Laplacian eigenvalues of G , respectively. In this paper we obtain μ i ( G ) ≤ i − 1 + min U i ⁡ max ⁡ { | N H ( v k ) ∪ N H ( v j ) | : v k v j ∈ E ( H ) } , where N H ( v k ) is the set of neighbors of vertex v k in V ( H ) = V ( G ) \ U i , U i is any ( i − 1 ) -subset of V ( G ) (here, we agree that i ∈ { 1 , … , n − 1 } and μ i ( G ) ≤ i − 1 if E ( H ) = ∅ ). For any graph G , this bound does not exceed the order of G . Moreover, we prove that max ⁡ { μ i ( G ) , q i ( G ) } ≤ max i ≤ k ≤ n ⁡ { d G ( v k ) + ∑ v j ∈ N G ( v k ) ∩ N d G ( v j ) d G ( v k ) } ≤ 2 d G ( v i ) , where d G ( v i ) is the i -th largest degree of G and N = { v i , v i + 1 , … , v n } . [ABSTRACT FROM AUTHOR]

Abstract: The remoteness of a connected graph is the maximum, over all vertices, of the average distance from a vertex to all others, while the proximity of a connected graph is the minimum, over all vertices, of the average distance from a vertex to all others. In this paper, we first deal with some conjectures on remoteness and proximity, among which two conjectures were proved, while the other two conjectures were disproved by counter examples. Then we obtain some new upper bounds for remoteness and proximity in terms of some graph invariants. Moreover, we use remoteness to give a new sufficient condition for a connected bipartite graph to be Hamiltonian. [Copyright &y& Elsevier]

Abstract: Let be a graph with vertices and edges. Also let be the eigenvalues of the Laplacian matrix of graph . The Laplacian energy of the graph is defined as In this paper, we present some lower and upper bounds for of graph in terms of , the number of edges and the maximum degree . Also we give a Nordhaus–Gaddum-type result for Laplacian energy of graphs. Moreover, we obtain a relation between Laplacian energy and Laplacian-energy-like invariant of graphs. [Copyright &y& Elsevier]

Abstract: Let be a simple graph with vertex set and edge set . The incidence matrix of G is the matrix whose -entry is 1 if is incident to and 0 otherwise. The incidence energy IE of G is the sum of the singular values of . In this paper we give lower and upper bounds for IE in terms of n, m, maximum degree, clique number, independence number, and the first Zagreb index. Moreover, we obtain Nordhaus–Gaddum-type results for IE. [Copyright &y& Elsevier]

Abstract: Let G be a connected graph of order n with Laplacian eigenvalues . The Laplacian-energy-like invariant of the graph G is defined as Lower and upper bounds for LEL are obtained, in terms of n, number of edges, maximum vertex degree, and number of spanning trees. [Copyright &y& Elsevier]

Abstract: Let be a simple graph with vertex set and edge set . The Laplacian matrix of G is , where is the diagonal matrix of its vertex degrees and is the adjacency matrix. Among all eigenvalues of the Laplacian matrix of a graph, the most studied is the second smallest, called the algebraic connectivity of a graph [18]. In this paper we give a lower bound on the algebraic connectivity of graphs. Moreover, we mention two conjectures, obtained by the system AutoGraphiX, about the algebraic connectivity , diameter and the minimum degree of graphs (see, [2], available online at http://www.gerad.ca/∼agx/):with equality if and only if G is isomorphic to a graph with and , and (ii) is minimum for a graph composed of 2 cliques on vertices with a common vertex if n is odd, and linked by an edge if n is even. Here we prove conjecture in (i) using the lower bound on the algebraic connectivity of graphs and conjecture in (ii), respectively. [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: Let be a simple connected graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of G is . In this paper, we obtain some new and improved sharp upper bounds on the spectral radius of the signless Laplacian matrix of a graph G. [Copyright &y& Elsevier]

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 a ( k ) = ( a 1 , a 2 , … , a k ) be a sequence of positive integers. A butterfly-like graph W p ( s ) ; a ( k ) is a graph consisting of s ( ≥ 1 ) cycle of lengths p + 1 , and k ( ≥ 1 ) paths P a 1 + 1 , P a 2 + 1 , …, P a k + 1 intersecting in a single vertex. The girth of a graph G is the length of a shortest cycle in G . Two graphs are said to be A - cospectral if they have the same adjacency spectrum. For a graph G , if there does not exist another non-isomorphic graph H such that G and H share the same Laplacian (respectively, signless Laplacian) spectrum, then we say that G is L − D S (respectively, Q − D S ). In this paper, we firstly prove that no two non-isomorphic butterfly-like graphs with the same girth are A -cospectral, and then present a new upper and lower bounds for the i -th largest eigenvalue of L ( G ) and Q ( G ) , respectively. By applying these new results, we give a positive answer to an open problem in Wen et al. (2015) [17] by proving that all the butterfly-like graphs W 2 ( s ) ; a ( k ) are both Q − D S and L − D S . [ABSTRACT FROM AUTHOR]