number of ordered triples (a, b, c) such that ab ∈ E(G) and bc / ∈ E(G). Observe that each vertex v has deg v = 0, 1, 2, 3, 4, or 5. If deg v is 0 or 5, then v is the second vertex in 0 such triples. If deg v is 1 or 4, then v is the second vertex in 4 such triples. If deg v is 2 or 3, then v is the second vertex in 6 such triples. Thus, there are at most 36 such ordered triples. Note that each...

For a graph G, we denote by σ2(G) the minimum degree sum of two non-adjacent vertices if G is non-complete; otherwise, σ2(G) = +∞. In this paper, we give the following two results; (i) If s1 and s2 are integers with s1, s2 ≥ 2 and if G is a non-complete graph with σ2(G) ≥ 2(s1 + s2 + 1) − 1, then G contains two vertexdisjoint subgraphs H1 and H2 such that each Hi is a graph of order at least si...

For a finite group G, the character degree graph ∆(G) is the graph whose vertices are the primes dividing the degrees of the ordinary irreducible characters of G, with distinct primes p and q joined by an edge if pq divides some character degree of G. We determine all graphs with four vertices that occur as ∆(G) for some nonsolvable group G. Along with previously known results on character degr...

A graph H is d-degenerate if every subgraph of it contains a vertex of degree smaller than d. For a graph G, let ad(G) denote the maximum number of vertices of an induced d-degenerate subgraph of G. Sharp lowers bounds for %(G) in terms of the degree sequence of G are obtained, and the minimum number of edges of a graph G with n vertices and ~2(G) < m is determined precisely for all m < n.