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This paper presents tight upper bounds for all of the signless Laplacian eigenvalues of a graph with prescribed order and minimum degree, which improve previously known upper bounds. Also, the relation between the number of signless Laplacian eigenvalues falling within specific intervals and various graph parameters such as independence, clique, chromatic, edge covering and matching numbers are explored. [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]

The singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E (A) is the sum of its singular values. The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E (A) ≥ 2 E (D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E (H) is an edge cut of G , then E (H) ≤ E (G) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E (G) ≥ 4 m + n (n − 2) | det ⁡ (A) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E (G) given in Oboudi (2019) [14]. [ABSTRACT FROM AUTHOR]

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ ( G ) ≥ 3 , then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D 1 is a primitive regular orientation and D 2 is a non-regular orientation of a given graph, then for sufficiently large t , the number of closed walks of length t in D 1 is more than the number of closed walks of length t in D 2 . [ABSTRACT FROM AUTHOR]

The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In Akbari and Hosseinzadeh (2020) [3] it was conjectured that for every graph G with maximum degree Δ (G) and minimum degree δ (G) whose adjacency matrix is non-singular, E (G) ≥ Δ (G) + δ (G) and the equality holds if and only if G is a complete graph. Let G be a connected graph with the edge set E (G). In this paper, first we show that E (L (G)) ≥ | E (G) | + Δ (G) − 5 , where L (G) denotes the line graph of G. Next, using this result, we prove the validity of the conjecture for the line of each connected graph of order at least 7. [ABSTRACT FROM AUTHOR]