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Bollobas and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $lambda^2_1(G)+lambda^2_2(G)leq frac{r-1}{r}cdot2m$, where $lambda_1(G)$ and $lambda_2(G)$ are the largest and the second largest eigenvalues of the adjacency matrix $A(G)$, respectively. In this paper, we confirm the conjecture in the case $r=2$, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to ErdH{o}s and Nosal respectively, we prove that every non-bipartite graph $G$ of order $n$ and size $m$ contains a triangle, if one of the following is true: (1) $lambda_1(G)geqsqrt{m-1}$ and $G eq C_5cup (n-5)K_1$; and (2) $lambda_1(G)geq lambda_1(S(K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil}))$ and $G eq S(K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil})$, where $S(K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil})$ is obtained from $K_{lfloorfrac{n-1}{2}rfloor,lceilfrac{n-1}{2}rceil}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.
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A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs with complex units, matrix properties are established. As with graphs and signed graphs, there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph.
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $delta^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $delta^c(G)geq frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $delta^c(G)geq frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k$ rainbow triangles, which can be seen as a generalization of their result. Li showed that for an edge-colored graph $G$ of order $n$, if $delta^c(G)geq frac{n+1}{2}$, then $G$ contains a rainbow triangle. We show that if $G$ is complete and $delta^c(G)geq frac{n}{2}$, then $G$ contains a rainbow triangle and the bound is sharp. Hu et al. showed that for an edge-colored graph $G$ of order $ngeq 20$, if $delta^c(G)geq frac{n+2}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. We show that if $G$ is complete with order $ngeq 8$ and $delta^c(G)geq frac{n+1}{2}$, then $G$ contains two vertex-disjoint rainbow triangles. Moreover, we improve the result of Hu et al. from $ngeq 20$ to $ngeq 7$, the best possible.
Let $G_1$ and $G_2$ be two simple connected graphs. The invariant textit{coronal} of graph is used in order to determine the $alpha$-eigenvalues of four different types of graph equations that are $G_1 circ G_2, G_1lozenge G_1$ and the other two`s are $G_1 odot G_2$ and $G_1 circleddash G_2$ which are obtained using the $R$-graph of $G_1$. As an application we construct infinitely many pairs of non-isomorphic $alpha$-Isospectral graph.
We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) frac{1}{3 sqrt 2} n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size.
For a real constant $alpha$, let $pi_3^alpha(G)$ be the minimum of twice the number of $K_2$s plus $alpha$ times the number of $K_3$s over all edge decompositions of $G$ into copies of $K_2$ and $K_3$, where $K_r$ denotes the complete graph on $r$ vertices. Let $pi_3^alpha(n)$ be the maximum of $pi_3^alpha(G)$ over all graphs $G$ with $n$ vertices. The extremal function $pi_3^3(n)$ was first studied by GyH{o}ri and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320]. In a recent progress on this problem, Kral, Lidicky, Martins and Pehova [Decomposing graphs into edges and triangles, Combin. Prob. Comput. 28 (2019) 465--472] proved via flag algebras that $pi_3^3(n)le (1/2+o(1))n^2$. We extend their result by determining the exact value of $pi_3^alpha(n)$ and the set of extremal graphs for all $alpha$ and sufficiently large $n$. In particular, we show for $alpha=3$ that $K_n$ and the complete bipartite graph $K_{lfloor n/2rfloor,lceil n/2rceil}$ are the only possible extremal examples for large $n$.
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