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Given a finite group G , the character graph , denoted by Δ (G) , for its irreducible character degrees is a graph with vertex set ρ (G) which is the set of prime numbers that divide the irreducible character degrees of G , and with { p , q } being an edge if there exists a non-linear χ ∈ Irr (G) whose degree is divisible by p q. In this paper, on one hand, we proceed by discussing the graphical shape of Δ (G) when it has cut vertices or small number of eigenvalues, and on the other hand we give some results on the group structure of G with such Δ (G). Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of character graphs of solvable groups with a cut vertex and diameter 3. Furthermore, we study solvable groups whose character graphs have at most two distinct eigenvalues. Moreover, we investigate the solvable groups whose character graphs are regular with three distinct eigenvalues. In addition, we give some lower bounds for the number of edges of Δ (G). [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]

A graph is called equimatchable if all of its maximal matchings have the same size. Kawarabayashi, Plummer, and Saito showed that the only connected equimatchable 3-regular graphs are K4 and K3, 3. We extend this result by showing that for an odd positive integer r, if G is a connected equimatchable r-regular graph, then [ABSTRACT FROM AUTHOR]

A Deza graph G with parameters (n , k , b , a) is a k -regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children G A and G B of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in G A or G B if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. [ABSTRACT FROM AUTHOR]