In this paper, we consider two typical problems on a locally finite connected graph. The first one is to study the Bochner formula for the Laplacian operator on a locally finite connected graph. The other one is to obtain global nontrivial nonnegative solution to porous-media equation via the use of Aronson-Benilan argument. We use the curvature dimension condition to give a characterization two point graph. We also give a porous-media equation criterion about stochastic completeness of the graph....

The Q-matrix of a connected graph = (V,E) is $Q={\left(q^{\partial (x,y)}\right)}_{x,y\in V}$, where ∂(x,y) is the graph distance. Let q() be the range of q ∈ (-1,1) for which the Q-matrix is strictly positive. We obtain a sufficient condition for the equality q(̃) = q() where ̃ is an extension of a finite graph by joining a square. Some concrete examples are discussed.

The power index of a square Boolean matrix A is the least integer d such that Ad is a linear combination of previous nonnegative powers of A. We determine the maximum power indices for the class of n×n primitive symmetric Boolean matrices of trace zero, the class of n×n irreducible nonprimitive symmetric Boolean matrices, and the class of n×n reducible symmetric Boolean matrices of trace zero, and characterize the extreme matrices respectively.

In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $\omega$$(2\le ...$

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H. Bart, A. P. M. Wagelmans (2000). The proof involves elements from integer programming and employs Farkas' lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred...

Let $G$ be a finite graph with an eigenvalue $\mu$ of multiplicity $m$. A set $X$ of $m$ vertices in $G$ is called a star set for $\mu$ in $G$ if $\mu$ is not an eigenvalue of the star complement $G\setminus X$ which is the subgraph of $G$ induced by vertices not in $X$. A vertex subset of a graph is $(\kappa ,\tau )$-regular if it induces a $\kappa$-regular subgraph and every vertex not in the subset has $\tau$ neighbors in it. We investigate the graphs having a $(\kappa ,\tau )$-regular set which induces a star complement for some eigenvalue. A survey of known results is provided...

Two inequalities for the Laplacian spread of graphs are proved in this note. These inequalities are reverse to those obtained by Z. You, B. Liu: The Laplacian spread of graphs, Czech. Math. J. 62 (2012), 155–168.

A graph is called distance integral (or $D$-integral) if all eigenvalues of its distance matrix are integers. In their study of $D$-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_1,p_2,p_3}$ with $p_1<p_2<p_3$, and $K_{p_1,p_2,p_3,p_4}$ with $p_1<p_2<p_3<p_4$, as well as the infinite classes of distance integral complete...

Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1,d_2,\cdots ,d_n)$, where $d_1\ge d_2\ge \cdots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho \left(G\right)$ and $\mu \left(G\right)$, respectively. We determine the graphs with $$\rho \left(G\right)=\frac{d_n-1}{2}+\sqrt{2m-n{d}_n+\frac{{(d_n+1)}^2}{4}}$$ and the graphs with $d_n\ge 1$ and $$\mu \left(G\right)=d_n+\frac{1}{2}+\sqrt{{\sum }_{i=1}^n{d}_i(d_i-d_n)+{\left(d_n-\frac{1}{2}\right)}^2}.$$ We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.

We give a novel upper bound on graph energy in terms of the vertex cover number, and present a complete characterization of the graphs whose energy equals twice their matching number.